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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G</font> being   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>  st (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> holds <br/> ex <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font>) ) ) &amp; <font color="Maroon" title="c1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  holds <br/> ex <font color="Olive" title="b2">g</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b2">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Olive" title="b1">G</font> &amp; <font color="Olive" title="b2">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b2">g</font> &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">g</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>) ) ) &amp; <font color="Maroon" title="c1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  implies  ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> ) )</span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">f</font> holds <br/> ex <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font>) ) ) &amp; <font color="Olive" title="b1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  holds <br/> ex <font color="Olive" title="b2">g</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b2">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b2">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Olive" title="b1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">g</font> );<br/>
<a NAME="E1:2"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>) ) ) &amp; <font color="Maroon" title="c4">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c4">f</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c4">f</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c4">f</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  implies  ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:2_1"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E3:2_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>) )
 <span class="kw">and </span><br/><a NAME="E4:2_1"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E5:2_1"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> 
 <span class="kw">and </span><br/><a NAME="E6:2_1"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> 
 <span class="kw">and </span><br/><a NAME="E7:2_1"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_1"/><span class="lab"><font color="Green" title="E11">A9</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> )</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">f</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c5">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c5">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c5">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c5">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c5">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g</font> )</span><br/><a NAME="E2:2_1_1_1"/><span class="lab"><font color="Green" title="E12">A10</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E1:2_1_1_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="finseq_1.html#T2" onmouseover="rs('finseq_1/T2')" onmouseout="rh()">FINSEQ_1:2</a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">g</font> holds <br/> for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j1</font>, <font color="Olive" title="b5">j2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">j1</font>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">i1</font>,<font color="Olive" title="b3">i2</font>) &amp; <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b4">j1</font>,<font color="Olive" title="b5">j2</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">j1</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">j2</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">g</font> &amp; <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">g</font> implies  for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font>) &amp; <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_1_1"/><span class="lab"><font color="Green" title="E13">A11</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">g</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_1_1"/><span class="lab"><font color="Green" title="E14">A12</font></span>: 
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font>) &amp; <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E3:2_1_1_1_1"/>
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="txt" href="#E1:2_1_1_1_1"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:2_1_1_1_1"/>
 for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">i2</font>, <font color="Olive" title="b3">j1</font>, <font color="Olive" title="b4">j2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">i2</font>) &amp; <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b3">j1</font>,<font color="Olive" title="b4">j2</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">j1</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="txt" href="#E2:2_1_1_1_1"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:2_1_1_1"/>
<font color="Maroon" title="c5">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="goboard1.html#D9" onmouseover="rs('goboard1/D9')" onmouseout="rh()">GOBOARD1:def 9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c5">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c5">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c5">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c5">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g</font> )</span><br/><span class="kw">thus </span><a NAME="E5:2_1_1_1"/>
( <font color="Maroon" title="c5">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c5">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c5">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c5">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c5">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2"/><span class="lab"><font color="Green" title="E11">A13</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> )</span><br/><div class="add"><a NAME="E2:2_1_1_2"/><span class="lab"><font color="Green" title="E12">A14</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c5">f1</font> = <font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font>;<br/><a NAME="E3:2_1_1_2"/><span class="lab"><font color="Green" title="E13">A15</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E1:2_1_1_2"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="goboard3.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><a NAME="E4:2_1_1_2"/><span class="lab"><font color="Green" title="E14">A16</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="goboard2.html#T7" onmouseover="rs('goboard2/T7')" onmouseout="rh()">GOBOARD2:7</a></span>;<br/><a NAME="E5:2_1_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D16" onmouseover="rs('finseq_1/D16')" onmouseout="rh()">FINSEQ_1:def 16</a></span>;<br/><a NAME="E6:2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A17</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="funct_1.html#T52" onmouseover="rs('funct_1/T52')" onmouseout="rh()">FUNCT_1:52</a></span>;<br/><a NAME="E7:2_1_1_2"/><span class="lab"><font color="Green" title="E16">A18</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E8:2_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E9:2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A19</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">j1</font>, <font color="Maroon" title="c7">j2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E11:2_1_1_2"/><span class="lab"><font color="Green" title="E18">A20</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c7">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E12:2_1_1_2"/><span class="lab"><font color="Green" title="E19">A21</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c7">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>;<br/><a NAME="E13:2_1_1_2"/><span class="lab"><font color="Green" title="E20">A22</font></span>: 
 <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D4" onmouseover="rs('matrix_0/D4')" onmouseout="rh()">MATRIX_0:def 4</a></span>;<br/><a NAME="E14:2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A23</font></span>: 
<font color="Maroon" title="c6">j1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/><a NAME="E15:2_1_1_2"/><span class="lab"><font color="Green" title="E22">A24</font></span>: 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E16:2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A25</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E17:2_1_1_2"/><span class="lab"><font color="Green" title="E24">A26</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E18:2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A27</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">i1</font>, <font color="Maroon" title="c9">i2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E20:2_1_1_2"/><span class="lab"><font color="Green" title="E26">A28</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E21:2_1_1_2"/><span class="lab"><font color="Green" title="E27">A29</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">l1</font> =  <a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c8">i1</font>), <font color="Maroon" title="c11">c1</font> =  <a href="matrix_0.html#K8" title="MATRIX_0:func.8">Col</a> (<font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c9">i2</font>) as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c12">x1</font> =  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">l1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c13">y1</font> =  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">l1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c14">x2</font> =  <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">c1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c15">y2</font> =  <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">c1</font>;<br/><a NAME="E23:2_1_1_2"/><span class="lab"><font color="Green" title="E28">A30</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">l1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><a NAME="E24:2_1_1_2"/><span class="lab"><font color="Green" title="E29">A31</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E25:2_1_1_2"/><span class="lab"><font color="Green" title="E30">A32</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topreal1.html#D5" onmouseover="rs('topreal1/D5')" onmouseout="rh()">TOPREAL1:def 5</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font> or  not <font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_1"/><span class="lab"><font color="Green" title="E31">A33</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_1"/><span class="lab"><font color="Green" title="E32">A34</font></span>: 
<font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<a NAME="E3:2_1_1_2_1"/>
<font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_1"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="txt" href="#E2:2_1_1_2_1"><span class="lab"><font color="Green" title="E32">A34</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E4:2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A35</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E24:2_1_1_2"><span class="lab"><font color="Green" title="E29">A31</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>;<br/>
<a NAME="E5:2_1_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E17:2_1_1_2"><span class="lab"><font color="Green" title="E24">A26</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/>
<a NAME="E6:2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A36</font></span>: 
<font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_1"><span class="lab"><font color="Green" title="E32">A34</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:2_1_1_2_1"/>
<font color="Maroon" title="c16">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_1"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="txt" href="#E2:2_1_1_2_1"><span class="lab"><font color="Green" title="E32">A34</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E8:2_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E24:2_1_1_2"><span class="lab"><font color="Green" title="E29">A31</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:2_1_1_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E1:2_1_1_2_1"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="txt" href="#E4:2_1_1_2_1"><span class="lab"><font color="Green" title="E33">A35</font></span></a>, <a class="txt" href="#E6:2_1_1_2_1"><span class="lab"><font color="Green" title="E34">A36</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) ) )</span><br/><span class="kw">assume </span><a NAME="E1:2_1_1_2_2"/><span class="lab"><font color="Green" title="E31">A37</font></span>: 
<font color="Maroon" title="c16">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) )</span><br/><a NAME="E2:2_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E24:2_1_1_2"><span class="lab"><font color="Green" title="E29">A31</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">i</font>, <font color="Maroon" title="c18">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:2_1_1_2_2"/><span class="lab"><font color="Green" title="E32">A38</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c18">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c19">i</font> = <font color="Maroon" title="c17">i</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i</font>,<font color="Olive" title="b1">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c19">i</font>,<font color="Olive" title="b1">j</font>) )</span><br/><span class="kw">take </span><font color="Maroon" title="c20">j</font> = <font color="Maroon" title="c18">j</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i</font>,<font color="Maroon" title="c20">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c19">i</font>,<font color="Maroon" title="c20">j</font>) )</span><br/><span class="kw">thus </span><a NAME="E5:2_1_1_2_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i</font>,<font color="Maroon" title="c20">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c19">i</font>,<font color="Maroon" title="c20">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E24:2_1_1_2"><span class="lab"><font color="Green" title="E29">A31</font></span></a>, <a class="txt" href="#E1:2_1_1_2_2"><span class="lab"><font color="Green" title="E31">A37</font></span></a>, <a class="txt" href="#E4:2_1_1_2_2"><span class="lab"><font color="Green" title="E32">A38</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">g1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E28:2_1_1_2"/><span class="lab"><font color="Green" title="E31">A39</font></span>: 
<font color="Maroon" title="c16">g1</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E29:2_1_1_2"/><span class="lab"><font color="Green" title="E32">A40</font></span>: 
<font color="Maroon" title="c16">g1</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E30:2_1_1_2"/><span class="lab"><font color="Green" title="E33">A41</font></span>: 
<font color="Maroon" title="c16">g1</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> 
 <span class="kw">and </span><br/><a NAME="E31:2_1_1_2"/><span class="lab"><font color="Green" title="E34">A42</font></span>: 
<font color="Maroon" title="c16">g1</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> 
 <span class="kw">and </span><br/><a NAME="E32:2_1_1_2"/><span class="lab"><font color="Green" title="E35">A43</font></span>: 
<font color="Maroon" title="c16">g1</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">and </span><br/><a NAME="E33:2_1_1_2"/><span class="lab"><font color="Green" title="E36">A44</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E34:2_1_1_2"/><span class="lab"><font color="Green" title="E37">A45</font></span>: 
<font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">and </span><br/><a NAME="E35:2_1_1_2"/><span class="lab"><font color="Green" title="E38">A46</font></span>: 
<font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E36:2_1_1_2"/><span class="lab"><font color="Green" title="E39">A47</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E6:2_1_1_2"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="txt" href="#E3:2_1_1_2"><span class="lab"><font color="Green" title="E13">A15</font></span></a>, <a class="txt" href="#E4:2_1_1_2"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="txt" href="#E25:2_1_1_2"><span class="lab"><font color="Green" title="E30">A32</font></span></a></span>;<br/><a NAME="E37:2_1_1_2"/><span class="lab"><font color="Green" title="E40">A48</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font> holds <br/> for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font>) &amp; <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">i</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">j</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:2_1_1_2"><span class="lab"><font color="Green" title="E31">A39</font></span></a>, <a class="ref" href="goboard1.html#D9" onmouseover="rs('goboard1/D9')" onmouseout="rh()">GOBOARD1:def 9</a></span>;<br/><a NAME="E38:2_1_1_2"/><span class="lab"><font color="Green" title="E41">A49</font></span>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> implies  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_1_1_2_3"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span><br/>

<a NAME="E2:2_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A50</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E3:2_1_1_2_3"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span><br/>

<a NAME="E4:2_1_1_2_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font> &amp; <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:2_1_1_2_3"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="txt" href="#E2:2_1_1_2_3"><span class="lab"><font color="Green" title="E42">A50</font></span></a>, <a class="ref" href="goboard1.html#T1" onmouseover="rs('goboard1/T1')" onmouseout="rh()">GOBOARD1:1</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E39:2_1_1_2"/><span class="lab"><font color="Green" title="E42">A51</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E40:2_1_1_2"/><span class="lab"><font color="Green" title="E43">A52</font></span>: 
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies (  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:2_1_1_2_4"/><span class="lab"><font color="Green" title="E44">A53</font></span>: 
<font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:2_1_1_2_4"/><span class="lab"><font color="Green" title="E45">A54</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">hence </span><a NAME="E3:2_1_1_2_4"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="ref" href="topreal1.html#T22" onmouseover="rs('topreal1/T22')" onmouseout="rh()">TOPREAL1:22</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E4:2_1_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A55</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#T22" onmouseover="rs('topreal1/T22')" onmouseout="rh()">TOPREAL1:22</a></span>;<br/>
<a NAME="E5:2_1_1_2_4"/><span class="lab"><font color="Green" title="E47">A56</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_1_2_4_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_4_1"/><span class="lab"><font color="Green" title="E48">A57</font></span>: 
<font color="Maroon" title="c18">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_4_1"/><span class="lab"><font color="Green" title="E49">A58</font></span>: 
<font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E5:2_1_1_2_4_1"/>
<font color="Maroon" title="c18">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_4_1"><span class="lab"><font color="Green" title="E48">A57</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:2_1_1_2_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_4"><span class="lab"><font color="Green" title="E46">A55</font></span></a>, <a class="ref" href="finseq_1.html#T2" onmouseover="rs('finseq_1/T2')" onmouseout="rh()">FINSEQ_1:2</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:2_1_1_2_4_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_4"><span class="lab"><font color="Green" title="E44">A53</font></span></a>, <a class="txt" href="#E4:2_1_1_2_4_1"><span class="lab"><font color="Green" title="E49">A58</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:2_1_1_2_4"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="txt" href="#E2:2_1_1_2_4"><span class="lab"><font color="Green" title="E45">A54</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/>
<a NAME="E7:2_1_1_2_4"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E8:2_1_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_4"><span class="lab"><font color="Green" title="E44">A53</font></span></a>, <a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E9:2_1_1_2_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:2_1_1_2_4"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_4"><span class="lab"><font color="Green" title="E47">A56</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E41:2_1_1_2"/><span class="lab"><font color="Green" title="E44">A59</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A60</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">c1</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
;<br/><a NAME="E43:2_1_1_2"/><span class="lab"><font color="Green" title="E46">A61</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">c1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><a NAME="E44:2_1_1_2"/><span class="lab"><font color="Green" title="E47">A62</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">l1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a>, <a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><a NAME="E45:2_1_1_2"/><span class="lab"><font color="Green" title="E48">A63</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E46:2_1_1_2"/><span class="lab"><font color="Green" title="E49">A64</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A65</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">c1</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>
;<br/><a NAME="E48:2_1_1_2"/><span class="lab"><font color="Green" title="E51">A66</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E13:2_1_1_2"><span class="lab"><font color="Green" title="E20">A22</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/><a NAME="E49:2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A67</font></span>: 
 <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">l1</font> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D4" onmouseover="rs('goboard1/D4')" onmouseout="rh()">GOBOARD1:def 4</a></span>;<br/><a NAME="E50:2_1_1_2"/><span class="lab"><font color="Green" title="E53">A68</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E13:2_1_1_2"><span class="lab"><font color="Green" title="E20">A22</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/><a NAME="E51:2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A69</font></span>: 
 <a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">c1</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard1.html#D7" onmouseover="rs('goboard1/D7')" onmouseout="rh()">GOBOARD1:def 7</a></span>;<br/><a NAME="E52:2_1_1_2"/><span class="lab"><font color="Green" title="E55">A70</font></span>: 
 <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">c1</font> is  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="ref" href="goboard1.html#D5" onmouseover="rs('goboard1/D5')" onmouseout="rh()">GOBOARD1:def 5</a></span>;<br/><a NAME="E53:2_1_1_2"/><span class="lab"><font color="Green" title="E56">A71</font></span>: 
 <a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">l1</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="ref" href="goboard1.html#D6" onmouseover="rs('goboard1/D6')" onmouseout="rh()">GOBOARD1:def 6</a></span>;<br/><a NAME="E54:2_1_1_2"/><span class="lab"><font color="Green" title="E57">A72</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>;<br/><a NAME="E55:2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A73</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E56:2_1_1_2"/><span class="lab"><font color="Green" title="E59">A74</font></span>: 
<font color="Maroon" title="c7">j2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E13:2_1_1_2"><span class="lab"><font color="Green" title="E20">A22</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/><a NAME="E57:2_1_1_2"/><span class="lab"><font color="Green" title="E60">A75</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font> holds <br/> ex <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:2_1_1_2"><span class="lab"><font color="Green" title="E31">A39</font></span></a>, <a class="ref" href="goboard1.html#D9" onmouseover="rs('goboard1/D9')" onmouseout="rh()">GOBOARD1:def 9</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1"/>
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j1</font> or <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E20:2_1_1_2"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E9:2_1_1_2"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E11:2_1_1_2"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E12:2_1_1_2"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="ref" href="goboard2.html#T11" onmouseover="rs('goboard2/T11')" onmouseout="rh()">GOBOARD2:11</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1"/><span class="lab"><font color="Green" title="E61">A76</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c17">ppi</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>);<br/><span class="kw">set </span><font color="Maroon" title="c18">pj</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>);<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">i2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">j2</font> &amp;  ex <font color="Olive" title="b1">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> ) ) or ( <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> &amp; contradiction ) or ( <font color="Maroon" title="c9">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j2</font> &amp;  ex <font color="Olive" title="b1">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> ) ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1"/>
( <font color="Maroon" title="c9">i2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">j2</font> or <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> or <font color="Maroon" title="c9">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A77</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">j2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> )</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1"/>
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E55:2_1_1_2"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A78</font></span>: 
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A79</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1"/>
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E55:2_1_1_2"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A80</font></span>: 
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A81</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A82</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E53:2_1_1_2"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A77</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E64">A79</font></span></a>, <a class="ref" href="seqm_3.html#D1" onmouseover="rs('seqm_3/D1')" onmouseout="rh()">SEQM_3:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">l</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">j2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A77</font></span></a>, <a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Olive" title="b1">m</font>);<br/><span class="kw">set </span><font color="Maroon" title="c20">lk</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E68">A83</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><div><span class="lab"><font color="Green" title="E69">A84</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> holds <br/>( <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> implies ( <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> ) )</span><br/><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_1_1"/>
<font color="Maroon" title="c21">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> )</span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A85</font></span>: 
<font color="Maroon" title="c21">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_1"/>
<font color="Maroon" title="c19">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T43" onmouseover="rs('xreal_1/T43')" onmouseout="rh()">XREAL_1:43</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">w</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E70">A85</font></span></a>, <a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_1"/>
( <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i2</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="ref" href="xreal_1.html#T43" onmouseover="rs('xreal_1/T43')" onmouseout="rh()">XREAL_1:43</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A86</font></span>: 
<font color="Maroon" title="c22">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E72">A87</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E70">A85</font></span></a>, <a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A87</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A86</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E11:2_1_1_2_7_1_1_1_1_1_1"/>
( <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_1_2"><span class="lab"><font color="Green" title="E20">A22</font></span></a>, <a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E70">A88</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> holds <br/> ex <font color="Olive" title="b2">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">p</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> implies  ex <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">n</font>,<font color="Olive" title="b1">p</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_1_2"/>
<font color="Maroon" title="c21">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">n</font>,<font color="Olive" title="b1">p</font>]</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">m</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c23">p</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c22">m</font>); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">n</font>,<font color="Maroon" title="c23">p</font>]</span><br/><span class="kw">thus </span><a NAME="E3:2_1_1_2_7_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">n</font>,<font color="Maroon" title="c23">p</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c21">g2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E14:2_1_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E71">A89</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_4.html#S1" onmouseover="rs('finseq_4/S1')" onmouseout="rh()">FINSEQ_4:sch 1</a>(<a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E70">A88</font></span></a>);<br/></span><span class="kw">take </span><font color="Maroon" title="c22">g</font> = <font color="Maroon" title="c16">g1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">g2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c22">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )</span><br/><a NAME="E15:2_1_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E72">A90</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> holds <br/> ex <font color="Olive" title="b2">k</font>, <font color="Olive" title="b3">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">k</font>,<font color="Olive" title="b3">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">k</font>,<font color="Olive" title="b3">m</font>) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> implies  ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font>) ) )</span><br/><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E73">A91</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font>) )</span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">m</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><span class="kw">take </span><font color="Maroon" title="c25">k</font> = <font color="Maroon" title="c8">i1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">k</font>,<font color="Olive" title="b1">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">k</font>,<font color="Olive" title="b1">m</font>) )</span><br/><span class="kw">take </span><font color="Maroon" title="c26">m</font> = <font color="Maroon" title="c24">m</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">k</font>,<font color="Maroon" title="c26">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">k</font>,<font color="Maroon" title="c26">m</font>) )</span><br/><span class="kw">thus </span><a NAME="E4:2_1_1_2_7_1_1_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">k</font>,<font color="Maroon" title="c26">m</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">k</font>,<font color="Maroon" title="c26">m</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E73">A91</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E17:2_1_1_2_7_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A92</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:2_1_1_2"><span class="lab"><font color="Green" title="E60">A75</font></span></a>, <a class="ref" href="goboard1.html#T23" onmouseover="rs('goboard1/T23')" onmouseout="rh()">GOBOARD1:23</a></span>;<br/><a NAME="E18:2_1_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E74">A93</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E19:2_1_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E75">A94</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E44:2_1_1_2"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E63">A78</font></span></a>, <a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><div><span class="lab"><font color="Green" title="E76">A95</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> holds <br/>( <font color="Olive" title="b2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b2">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> &amp; <font color="Olive" title="b2">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> holds <br/>( <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c24">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p</font> implies ( <font color="Maroon" title="c24">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c24">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c24">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> &amp; <font color="Maroon" title="c24">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E77">A96</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E78">A97</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c24">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c24">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> &amp; <font color="Maroon" title="c24">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c25">n1</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E77">A96</font></span></a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_4"/>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E77">A96</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A98</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c26">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c25">n1</font>);<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E80">A99</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c25">n1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E74">A93</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E77">A96</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E81">A100</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E74">A93</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E77">A96</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A101</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E49:2_1_1_2"><span class="lab"><font color="Green" title="E52">A67</font></span></a>, <a class="txt" href="#E44:2_1_1_2"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="ref" href="seqm_3.html#D10" onmouseover="rs('seqm_3/D10')" onmouseout="rh()">SEQM_3:def 10</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_1_4"/>
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:2_1_1_2"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E81">A100</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A102</font></span>: 
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c25">n1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E81">A100</font></span></a>, <a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A103</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c25">n1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E81">A100</font></span></a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_1_1_1_1_4"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c25">n1</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:2_1_1_2"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E81">A100</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E83">A102</font></span></a>, <a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E13:2_1_1_2_7_1_1_1_1_1_4"/>
( <font color="Maroon" title="c24">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c24">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E53:2_1_1_2"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E64">A79</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E66">A81</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E75">A94</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E78">A97</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E81">A100</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E80">A99</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E79">A98</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E82">A101</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E84">A103</font></span></a>, <a class="ref" href="seq_4.html#T137" onmouseover="rs('seq_4/T137')" onmouseout="rh()">SEQ_4:137</a>, <a class="ref" href="xreal_1.html#T43" onmouseover="rs('xreal_1/T43')" onmouseout="rh()">XREAL_1:43</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> &amp; <font color="Maroon" title="c24">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><a NAME="E14:2_1_1_2_7_1_1_1_1_1_4"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">l1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E15:2_1_1_2_7_1_1_1_1_1_4"/>
<font color="Maroon" title="c24">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E78">A97</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E81">A100</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E80">A99</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E83">A102</font></span></a>, <a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/><a NAME="E16:2_1_1_2_7_1_1_1_1_1_4"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E77">A96</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E17:2_1_1_2_7_1_1_1_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c25">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T44" onmouseover="rs('xreal_1/T44')" onmouseout="rh()">XREAL_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E18:2_1_1_2_7_1_1_1_1_1_4"/>
<font color="Maroon" title="c24">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E53:2_1_1_2"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E64">A79</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E78">A97</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E81">A100</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E80">A99</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E84">A103</font></span></a>, <a class="ref" href="seqm_3.html#D1" onmouseover="rs('seqm_3/D1')" onmouseout="rh()">SEQM_3:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E21:2_1_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E77">A104</font></span>: 
<font color="Maroon" title="c21">g2</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topreal1.html#D5" onmouseover="rs('topreal1/D5')" onmouseout="rh()">TOPREAL1:def 5</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> or  not <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c24">p</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>;<br/>
<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_1_5"/><span class="lab"><font color="Green" title="E78">A105</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<a NAME="E2:2_1_1_2_7_1_1_1_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E3:2_1_1_2_7_1_1_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A106</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a></span>;<br/>
<a NAME="E4:2_1_1_2_7_1_1_1_1_1_5"/>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_5"><span class="lab"><font color="Green" title="E78">A105</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_1_1_1_1_5"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_5"><span class="lab"><font color="Green" title="E79">A106</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E78">A107</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_6"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q</font> holds <br/><font color="Olive" title="b4">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> holds <br/><font color="Olive" title="b2">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c25">p</font>, <font color="Maroon" title="c26">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">p</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q</font> implies <font color="Maroon" title="c26">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_1_6"/><span class="lab"><font color="Green" title="E79">A108</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_6"/><span class="lab"><font color="Green" title="E80">A109</font></span>: 
<font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_6"/><span class="lab"><font color="Green" title="E81">A110</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_6"/><span class="lab"><font color="Green" title="E82">A111</font></span>: 
( <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">p</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_6"/><span class="lab"><font color="Green" title="E83">A112</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E79">A108</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c27">n1</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c28">m1</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E79">A108</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E80">A109</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c29">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c27">n1</font>);<br/><span class="kw">set </span><font color="Maroon" title="c30">pm</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c28">m1</font>);<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_6"/><span class="lab"><font color="Green" title="E84">A113</font></span>: 
<font color="Maroon" title="c28">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E81">A110</font></span></a>, <a class="ref" href="xreal_1.html#T15" onmouseover="rs('xreal_1/T15')" onmouseout="rh()">XREAL_1:15</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_6"/>
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:2_1_1_2"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E79">A108</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c27">n1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E83">A112</font></span></a>, <a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A114</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c27">n1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E83">A112</font></span></a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_1_6"/><span class="lab"><font color="Green" title="E86">A115</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E80">A109</font></span></a></span>;<br/><a NAME="E12:2_1_1_2_7_1_1_1_1_1_6"/>
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:2_1_1_2"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E80">A109</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E13:2_1_1_2_7_1_1_1_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c28">m1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E86">A115</font></span></a>, <a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>;<br/><a NAME="E14:2_1_1_2_7_1_1_1_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A116</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c28">m1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E86">A115</font></span></a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><a NAME="E15:2_1_1_2_7_1_1_1_1_1_6"/>
( <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c27">n1</font>) &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c28">m1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E79">A108</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E80">A109</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E16:2_1_1_2_7_1_1_1_1_1_6"/>
<font color="Maroon" title="c26">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:2_1_1_2"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E82">A111</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E83">A112</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E86">A115</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E84">A113</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E85">A114</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1_6"><span class="lab"><font color="Green" title="E87">A116</font></span></a>, <a class="ref" href="seqm_3.html#D1" onmouseover="rs('seqm_3/D1')" onmouseout="rh()">SEQM_3:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E23:2_1_1_2_7_1_1_1_1_1"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> holds <br/> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b2">m</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> implies  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_1_7"/><span class="lab"><font color="Green" title="E79">A117</font></span>: 
<font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_7"/><span class="lab"><font color="Green" title="E80">A118</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_7"/><span class="lab"><font color="Green" title="E81">A119</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_7"/><span class="lab"><font color="Green" title="E82">A120</font></span>: 
<font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_7"/><span class="lab"><font color="Green" title="E83">A121</font></span>: 
<font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_7"/><span class="lab"><font color="Green" title="E84">A122</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D7" onmouseover="rs('xboole_0/D7')" onmouseout="rh()">XBOOLE_0:def 7</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c25">p1</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c26">p2</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <font color="Maroon" title="c27">q1</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font>, <font color="Maroon" title="c28">q2</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E8:2_1_1_2_7_1_1_1_1_1_7"/><span class="lab"><font color="Green" title="E85">A123</font></span>: 
( <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E80">A118</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E81">A119</font></span></a></span>;<br/>
<a NAME="E9:2_1_1_2_7_1_1_1_1_1_7"/>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E10:2_1_1_2_7_1_1_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A124</font></span>: 
<font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A107</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E80">A118</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E81">A119</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c29">lp</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c30">lq</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/>
<a NAME="E11:2_1_1_2_7_1_1_1_1_1_7"/><span class="lab"><font color="Green" title="E87">A125</font></span>: 
( <font color="Maroon" title="c25">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c26">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_1_1_1_1_7"/>
<font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_1_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A126</font></span>: 
<font color="Maroon" title="c28">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A107</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E82">A120</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E83">A121</font></span></a></span>;<br/>
<a NAME="E14:2_1_1_2_7_1_1_1_1_1_7"/><span class="lab"><font color="Green" title="E89">A127</font></span>: 
( <font color="Maroon" title="c27">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c28">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c31">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span>;<br/>
<a NAME="E15:2_1_1_2_7_1_1_1_1_1_7"/><span class="lab"><font color="Green" title="E90">A128</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E84">A122</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E16:2_1_1_2_7_1_1_1_1_1_7"/><span class="lab"><font color="Green" title="E91">A129</font></span>: 
( <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E82">A120</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E83">A121</font></span></a></span>;<br/>
<a NAME="E17:2_1_1_2_7_1_1_1_1_1_7"/><span class="lab"><font color="Green" title="E92">A130</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E84">A122</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E18:2_1_1_2_7_1_1_1_1_1_7"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E82">A120</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E83">A121</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c28">q2</font>,<font color="Maroon" title="c27">q1</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E88">A126</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E91">A129</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E89">A127</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>
;<br/>
<a NAME="E20:2_1_1_2_7_1_1_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A131</font></span>: 
 ex <font color="Olive" title="b1">tm</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">tm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> &amp; <font color="Olive" title="b1">tm</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">tm</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">tm</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E92">A130</font></span></a></span>;<br/>
<a NAME="E21:2_1_1_2_7_1_1_1_1_1_7"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E80">A118</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E81">A119</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c26">p2</font>,<font color="Maroon" title="c25">p1</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E86">A124</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E85">A123</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E87">A125</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>
;<br/>
<a NAME="E23:2_1_1_2_7_1_1_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A132</font></span>: 
 ex <font color="Olive" title="b1">tn</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">tn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> &amp; <font color="Olive" title="b1">tn</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">tn</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">tn</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E90">A128</font></span></a></span>;<br/>
<a NAME="E24:2_1_1_2_7_1_1_1_1_1_7"/>
<font color="Maroon" title="c27">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A107</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E79">A117</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E81">A119</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E82">A120</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:2_1_1_2_7_1_1_1_1_1_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E94">A132</font></span></a>, <a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1_7"><span class="lab"><font color="Green" title="E93">A131</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E24:2_1_1_2_7_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A133</font></span>: 
<font color="Maroon" title="c21">g2</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#T1" onmouseover="rs('goboard2/T1')" onmouseout="rh()">GOBOARD2:1</a></span>;<br/><a NAME="E25:2_1_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E80">A134</font></span>: 
 not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c23">ls</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> ) </span> } </span> ;<br/>
<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_1_8"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_8"/><span class="lab"><font color="Green" title="E81">A135</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c24">X</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_8"/><span class="lab"><font color="Green" title="E82">A136</font></span>: 
<font color="Maroon" title="c24">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_8"/><span class="lab"><font color="Green" title="E83">A137</font></span>: 
<font color="Maroon" title="c24">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c25">m</font>)
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_8"/><span class="lab"><font color="Green" title="E84">A138</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">m</font> &amp; <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_8"><span class="lab"><font color="Green" title="E82">A136</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">q1</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">m</font>, <font color="Maroon" title="c27">q2</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E9:2_1_1_2_7_1_1_1_1_1_8"/><span class="lab"><font color="Green" title="E85">A139</font></span>: 
<font color="Maroon" title="c25">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_8"><span class="lab"><font color="Green" title="E84">A138</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E10:2_1_1_2_7_1_1_1_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A140</font></span>: 
<font color="Maroon" title="c26">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c28">lq</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/>
<a NAME="E11:2_1_1_2_7_1_1_1_1_1_8"/><span class="lab"><font color="Green" title="E87">A141</font></span>: 
( <font color="Maroon" title="c26">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c27">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_1_1_1_1_8"/><span class="lab"><font color="Green" title="E88">A142</font></span>: 
<font color="Maroon" title="c25">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_8"><span class="lab"><font color="Green" title="E84">A138</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_1_1_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A143</font></span>: 
<font color="Maroon" title="c27">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a></span>;<br/>
<a NAME="E14:2_1_1_2_7_1_1_1_1_1_8"/>
<font color="Maroon" title="c25">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E15:2_1_1_2_7_1_1_1_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A144</font></span>: 
<font color="Maroon" title="c27">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A107</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_1_8"><span class="lab"><font color="Green" title="E85">A139</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_1_8"><span class="lab"><font color="Green" title="E88">A142</font></span></a></span>;<br/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c25">m</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c27">q2</font>,<font color="Maroon" title="c26">q1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_8"><span class="lab"><font color="Green" title="E84">A138</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_8"><span class="lab"><font color="Green" title="E86">A140</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_1_8"><span class="lab"><font color="Green" title="E89">A143</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1_8"><span class="lab"><font color="Green" title="E90">A144</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_8"><span class="lab"><font color="Green" title="E87">A141</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>
;<br/>
<a NAME="E17:2_1_1_2_7_1_1_1_1_1_8"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_8"><span class="lab"><font color="Green" title="E81">A135</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_8"><span class="lab"><font color="Green" title="E83">A137</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:2_1_1_2_7_1_1_1_1_1_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_1_8"><span class="lab"><font color="Green" title="E85">A139</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E26:2_1_1_2_7_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E44:2_1_1_2"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E65">A80</font></span></a>, <a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><a NAME="E27:2_1_1_2_7_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A145</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E49:2_1_1_2"><span class="lab"><font color="Green" title="E52">A67</font></span></a>, <a class="txt" href="#E44:2_1_1_2"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E75">A94</font></span></a>, <a class="ref" href="seqm_3.html#D10" onmouseover="rs('seqm_3/D10')" onmouseout="rh()">SEQM_3:def 10</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_9"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">m</font> holds <br/> not <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">m</font> implies  not <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_1_9"/><span class="lab"><font color="Green" title="E82">A146</font></span>: 
( <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> )
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_9"/><span class="lab"><font color="Green" title="E83">A147</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c25">n1</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c26">m1</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_9"><span class="lab"><font color="Green" title="E82">A146</font></span></a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_9"/><span class="lab"><font color="Green" title="E84">A148</font></span>: 
( <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c25">n1</font>) &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c26">m1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_9"><span class="lab"><font color="Green" title="E82">A146</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E5:2_1_1_2_7_1_1_1_1_1_9"/><span class="lab"><font color="Green" title="E85">A149</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_9"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_9"><span class="lab"><font color="Green" title="E82">A146</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_9"/><span class="kw">then </span>
<font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_9"><span class="lab"><font color="Green" title="E84">A148</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_9"><span class="lab"><font color="Green" title="E85">A149</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E8:2_1_1_2_7_1_1_1_1_1_9"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_9"><span class="lab"><font color="Green" title="E83">A147</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E29:2_1_1_2_7_1_1_1_1_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font>
 ;<br/><a NAME="E30:2_1_1_2_7_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A150</font></span>: 
<font color="Maroon" title="c21">g2</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T9" onmouseover="rs('partfun2/T9')" onmouseout="rh()">PARTFUN2:9</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">m1</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">l</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><a NAME="E32:2_1_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E83">A151</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><span class="lab"><font color="Green" title="E84">A152</font></span>:  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E12:2_1_1_2"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E67">A82</font></span></a>, <a class="txt" href="#E27:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E81">A145</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A83</font></span></a>, <a class="txt" href="#E32:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E83">A151</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>
;<br/><a NAME="E34:2_1_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E85">A153</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font> or <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_1_11"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_11"/><span class="lab"><font color="Green" title="E86">A154</font></span>: 
<font color="Maroon" title="c25">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_11"/><span class="lab"><font color="Green" title="E87">A155</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">n1</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c25">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E74">A93</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_11"><span class="lab"><font color="Green" title="E86">A154</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c27">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c26">n1</font>);<br/>
<a NAME="E6:2_1_1_2_7_1_1_1_1_1_11"/><span class="lab"><font color="Green" title="E88">A156</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c26">n1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E74">A93</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_11"><span class="lab"><font color="Green" title="E86">A154</font></span></a></span>;<br/>
<a NAME="E7:2_1_1_2_7_1_1_1_1_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A157</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c26">n1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_11"><span class="lab"><font color="Green" title="E86">A154</font></span></a></span>;<br/>
<a NAME="E8:2_1_1_2_7_1_1_1_1_1_11"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c26">n1</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c26">n1</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_11"><span class="lab"><font color="Green" title="E86">A154</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_11"><span class="lab"><font color="Green" title="E88">A156</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_1_1_1_1_11"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A152</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_11"><span class="lab"><font color="Green" title="E87">A155</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_11"><span class="lab"><font color="Green" title="E88">A156</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_11"><span class="lab"><font color="Green" title="E89">A157</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E86">A158</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_12"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> holds <br/> for <font color="Olive" title="b2">l1</font>, <font color="Olive" title="b3">l2</font>, <font color="Olive" title="b4">l3</font>, <font color="Olive" title="b5">l4</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">l1</font>,<font color="Olive" title="b3">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">l3</font>,<font color="Olive" title="b5">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">l1</font>,<font color="Olive" title="b3">l2</font>) &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b4">l3</font>,<font color="Olive" title="b5">l4</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> implies  for <font color="Olive" title="b1">l1</font>, <font color="Olive" title="b2">l2</font>, <font color="Olive" title="b3">l3</font>, <font color="Olive" title="b4">l4</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font>) &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_1_12"/><span class="lab"><font color="Green" title="E87">A159</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_12"/><span class="lab"><font color="Green" title="E88">A160</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">l1</font>, <font color="Olive" title="b2">l2</font>, <font color="Olive" title="b3">l3</font>, <font color="Olive" title="b4">l4</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font>) &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c25">m1</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">n</font>, <font color="Maroon" title="c26">m2</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_12"><span class="lab"><font color="Green" title="E87">A159</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_12"><span class="lab"><font color="Green" title="E88">A160</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c27">l1</font>, <font color="Maroon" title="c28">l2</font>, <font color="Maroon" title="c29">l3</font>, <font color="Maroon" title="c30">l4</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">l1</font>,<font color="Maroon" title="c28">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">l3</font>,<font color="Maroon" title="c30">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c27">l1</font>,<font color="Maroon" title="c28">l2</font>) &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c29">l3</font>,<font color="Maroon" title="c30">l4</font>) implies <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c29">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c30">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_12"/><span class="lab"><font color="Green" title="E89">A161</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">l1</font>,<font color="Maroon" title="c28">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_12"/><span class="lab"><font color="Green" title="E90">A162</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">l3</font>,<font color="Maroon" title="c30">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_12"/><span class="lab"><font color="Green" title="E91">A163</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c27">l1</font>,<font color="Maroon" title="c28">l2</font>)
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_12"/><span class="lab"><font color="Green" title="E92">A164</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c29">l3</font>,<font color="Maroon" title="c30">l4</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c29">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c30">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_12"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c26">m2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_12"><span class="lab"><font color="Green" title="E88">A160</font></span></a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_1_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A165</font></span>: 
( <font color="Maroon" title="c29">l3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> &amp; <font color="Maroon" title="c30">l4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">m2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_12"><span class="lab"><font color="Green" title="E90">A162</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_12"><span class="lab"><font color="Green" title="E92">A164</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1_12"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c25">m1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_12"><span class="lab"><font color="Green" title="E87">A159</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_1_12"/><span class="kw">then </span>
( <font color="Maroon" title="c27">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> &amp; <font color="Maroon" title="c28">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">m1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_12"><span class="lab"><font color="Green" title="E89">A161</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_12"><span class="lab"><font color="Green" title="E91">A163</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c29">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c30">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> = 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_1_12"><span class="lab"><font color="Green" title="E93">A165</font></span></a>, <a class="ref" href="absvalue.html#T2" onmouseover="rs('absvalue/T2')" onmouseout="rh()">ABSVALUE:2</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_13"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">l1</font>, <font color="Olive" title="b2">l2</font>, <font color="Olive" title="b3">l3</font>, <font color="Olive" title="b4">l4</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font>) &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font>) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font> &amp; 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">l1</font>, <font color="Maroon" title="c25">l2</font>, <font color="Maroon" title="c26">l3</font>, <font color="Maroon" title="c27">l4</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">l1</font>,<font color="Maroon" title="c25">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">l3</font>,<font color="Maroon" title="c27">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c24">l1</font>,<font color="Maroon" title="c25">l2</font>) &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c26">l3</font>,<font color="Maroon" title="c27">l4</font>) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font> &amp; 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> implies <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_1_13"/><span class="lab"><font color="Green" title="E87">A166</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">l1</font>,<font color="Maroon" title="c25">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_13"/><span class="lab"><font color="Green" title="E88">A167</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">l3</font>,<font color="Maroon" title="c27">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_13"/><span class="lab"><font color="Green" title="E89">A168</font></span>: 
<font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c24">l1</font>,<font color="Maroon" title="c25">l2</font>)
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_13"/><span class="lab"><font color="Green" title="E90">A169</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c26">l3</font>,<font color="Maroon" title="c27">l4</font>)
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_13"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_13"/><span class="lab"><font color="Green" title="E91">A170</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c28">m1</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_13"><span class="lab"><font color="Green" title="E91">A170</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_13"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c28">m1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_13"><span class="lab"><font color="Green" title="E91">A170</font></span></a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_1_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A171</font></span>: 
( <font color="Maroon" title="c26">l3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> &amp; <font color="Maroon" title="c27">l4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">m1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_13"><span class="lab"><font color="Green" title="E88">A167</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_13"><span class="lab"><font color="Green" title="E90">A169</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1_13"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_1_13"/><span class="kw">then </span>
( <font color="Maroon" title="c24">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> &amp; <font color="Maroon" title="c25">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E20:2_1_1_2"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_13"><span class="lab"><font color="Green" title="E87">A166</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_13"><span class="lab"><font color="Green" title="E89">A168</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> = 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_1_13"><span class="lab"><font color="Green" title="E92">A171</font></span></a>, <a class="ref" href="absvalue.html#T2" onmouseover="rs('absvalue/T2')" onmouseout="rh()">ABSVALUE:2</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E37:2_1_1_2_7_1_1_1_1_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> holds <br/> for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font>) &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">i</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">j</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:2_1_1_2"><span class="lab"><font color="Green" title="E40">A48</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E86">A158</font></span></a>, <a class="ref" href="goboard1.html#T24" onmouseover="rs('goboard1/T24')" onmouseout="rh()">GOBOARD1:24</a></span>;<br/><span class="kw">hence </span><a NAME="E38:2_1_1_2_7_1_1_1_1_1"/>
<font color="Maroon" title="c22">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E73">A92</font></span></a>, <a class="ref" href="goboard1.html#D9" onmouseover="rs('goboard1/D9')" onmouseout="rh()">GOBOARD1:def 9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )</span><br/><a NAME="E39:2_1_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E87">A172</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E12:2_1_1_2"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E40:2_1_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E88">A173</font></span>: 
 not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_14"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_1_14"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_14"/><span class="lab"><font color="Green" title="E89">A174</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_14"/><span class="lab"><font color="Green" title="E90">A175</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c25">n1</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E74">A93</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_14"><span class="lab"><font color="Green" title="E89">A174</font></span></a></span>;<br/>
<a NAME="E6:2_1_1_2_7_1_1_1_1_1_14"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c25">n1</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E74">A93</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_14"><span class="lab"><font color="Green" title="E89">A174</font></span></a></span>;<br/>
<a NAME="E7:2_1_1_2_7_1_1_1_1_1_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A176</font></span>: 
<font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_14"><span class="lab"><font color="Green" title="E90">A175</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/>
<a NAME="E8:2_1_1_2_7_1_1_1_1_1_14"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E74">A93</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_14"><span class="lab"><font color="Green" title="E89">A174</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_1_1_1_1_14"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_14"><span class="lab"><font color="Green" title="E91">A176</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E41:2_1_1_2_7_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_15"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c24">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_1_15"/><span class="lab"><font color="Green" title="E89">A177</font></span>: 
 not <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:2_1_1_2_7_1_1_1_1_1_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A178</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:2_1_1_2_7_1_1_1_1_1_15"/><span class="lab"><font color="Green" title="E91">A179</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_15"><span class="lab"><font color="Green" title="E89">A177</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_15_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_15_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_1_15_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_15_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_15_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_1_2_7_1_1_1_1_1_15_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E40:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E88">A173</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_15"><span class="lab"><font color="Green" title="E91">A179</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_15"><span class="lab"><font color="Green" title="E90">A178</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_15_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_15_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_15_1_1_2"/><span class="kw">then </span>
(  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E38:2_1_1_2"><span class="lab"><font color="Green" title="E41">A49</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E85">A153</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_15"><span class="lab"><font color="Green" title="E91">A179</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_15"><span class="lab"><font color="Green" title="E90">A178</font></span></a>, <a class="ref" href="goboard2.html#T4" onmouseover="rs('goboard2/T4')" onmouseout="rh()">GOBOARD2:4</a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_1_1_1_1_15_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E88">A173</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_15"><span class="lab"><font color="Green" title="E90">A178</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_1_1_1_1_15"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E42:2_1_1_2_7_1_1_1_1_1"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 ;<br/><span class="kw">hence </span><a NAME="E43:2_1_1_2_7_1_1_1_1_1"/>
<font color="Maroon" title="c22">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:2_1_1_2"><span class="lab"><font color="Green" title="E32">A40</font></span></a>, <a class="txt" href="#E30:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E82">A150</font></span></a>, <a class="ref" href="finseq_3.html#T91" onmouseover="rs('finseq_3/T91')" onmouseout="rh()">FINSEQ_3:91</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )</span><br/><a NAME="E44:2_1_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E89">A180</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> holds <br/><span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_16"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> &amp; <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> implies <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_1_16"/><span class="lab"><font color="Green" title="E90">A181</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_16"/><span class="lab"><font color="Green" title="E91">A182</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E3:2_1_1_2_7_1_1_1_1_1_16"/><span class="lab"><font color="Green" title="E92">A183</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E90">A181</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E91">A182</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/>
<a NAME="E4:2_1_1_2_7_1_1_1_1_1_16"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E5:2_1_1_2_7_1_1_1_1_1_16"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A152</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E85">A153</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_16"/><span class="lab"><font color="Green" title="E93">A184</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u1</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_16"/><span class="lab"><font color="Green" title="E94">A185</font></span>: 
<font color="Maroon" title="c25">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E9:2_1_1_2_7_1_1_1_1_1_16"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1_16"/>
<font color="Maroon" title="c25">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/>
<a NAME="E11:2_1_1_2_7_1_1_1_1_1_16"/><span class="lab"><font color="Green" title="E95">A186</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E90">A181</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E91">A182</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_1_1_1_1_16"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_1_1_1_1_16"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A152</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E85">A153</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">u2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E15:2_1_1_2_7_1_1_1_1_1_16"/><span class="lab"><font color="Green" title="E96">A187</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u2</font>
 <span class="kw">and </span><br/><a NAME="E16:2_1_1_2_7_1_1_1_1_1_16"/><span class="lab"><font color="Green" title="E97">A188</font></span>: 
<font color="Maroon" title="c26">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E17:2_1_1_2_7_1_1_1_1_1_16"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E18:2_1_1_2_7_1_1_1_1_1_16"/>
<font color="Maroon" title="c26">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/>
<a NAME="E19:2_1_1_2_7_1_1_1_1_1_16"/>
( <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E20:2_1_1_2_7_1_1_1_1_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A189</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c25">u1</font>,<font color="Maroon" title="c26">u2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E91">A182</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E93">A184</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E96">A187</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E21:2_1_1_2_7_1_1_1_1_1_16"/>
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E22:2_1_1_2_7_1_1_1_1_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A190</font></span>: 
<font color="Maroon" title="c26">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A107</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E92">A183</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E95">A186</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E93">A184</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E96">A187</font></span></a></span>;<br/>
<a NAME="E23:2_1_1_2_7_1_1_1_1_1_16"/><span class="lab"><font color="Green" title="E100">A191</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E90">A181</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E91">A182</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/>
<a NAME="E24:2_1_1_2_7_1_1_1_1_1_16"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E25:2_1_1_2_7_1_1_1_1_1_16"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A152</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E85">A153</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E27:2_1_1_2_7_1_1_1_1_1_16"/><span class="lab"><font color="Green" title="E101">A192</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">u</font>
 <span class="kw">and </span><br/><a NAME="E28:2_1_1_2_7_1_1_1_1_1_16"/><span class="lab"><font color="Green" title="E102">A193</font></span>: 
<font color="Maroon" title="c27">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E29:2_1_1_2_7_1_1_1_1_1_16"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E30:2_1_1_2_7_1_1_1_1_1_16"/>
<font color="Maroon" title="c27">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/>
<a NAME="E31:2_1_1_2_7_1_1_1_1_1_16"/>
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E32:2_1_1_2_7_1_1_1_1_1_16"/><span class="kw">then </span>
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E91">A182</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E33:2_1_1_2_7_1_1_1_1_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A194</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c27">u</font>,<font color="Maroon" title="c25">u1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E90">A181</font></span></a>, <a class="txt" href="#E27:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E101">A192</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E93">A184</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c28">lg</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/>
<a NAME="E34:2_1_1_2_7_1_1_1_1_1_16"/>
<font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E35:2_1_1_2_7_1_1_1_1_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A195</font></span>: 
<font color="Maroon" title="c25">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A107</font></span></a>, <a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E100">A191</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E92">A183</font></span></a>, <a class="txt" href="#E27:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E101">A192</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E93">A184</font></span></a></span>;<br/>
<a NAME="E36:2_1_1_2_7_1_1_1_1_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A196</font></span>: 
<font color="Maroon" title="c25">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E94">A185</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E99">A190</font></span></a></span>;<br/>
<a NAME="E37:2_1_1_2_7_1_1_1_1_1_16"/>
( <font color="Maroon" title="c27">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c26">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<a NAME="E38:2_1_1_2_7_1_1_1_1_1_16"/><span class="kw">then </span>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E101">A192</font></span></a>, <a class="txt" href="#E28:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E102">A193</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E96">A187</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E97">A188</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E99">A190</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E104">A195</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E39:2_1_1_2_7_1_1_1_1_1_16"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E101">A192</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E93">A184</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E96">A187</font></span></a>, <a class="txt" href="#E33:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E103">A194</font></span></a>, <a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E98">A189</font></span></a>, <a class="txt" href="#E36:2_1_1_2_7_1_1_1_1_1_16"><span class="lab"><font color="Green" title="E105">A196</font></span></a>, <a class="ref" href="topreal1.html#T8" onmouseover="rs('topreal1/T8')" onmouseout="rh()">TOPREAL1:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E45:2_1_1_2_7_1_1_1_1_1"/>
<font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_17"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topreal1.html#D6" onmouseover="rs('topreal1/D6')" onmouseout="rh()">TOPREAL1:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> or  not <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> or <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_1_17"/><span class="lab"><font color="Green" title="E90">A197</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_17"/><span class="lab"><font color="Green" title="E91">A198</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E3:2_1_1_2_7_1_1_1_1_1_17"/><span class="lab"><font color="Green" title="E92">A199</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E91">A198</font></span></a></span>;<br/>
<a NAME="E4:2_1_1_2_7_1_1_1_1_1_17"/><span class="lab"><font color="Green" title="E93">A200</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<a NAME="E5:2_1_1_2_7_1_1_1_1_1_17"/><span class="lab"><font color="Green" title="E94">A201</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E6:2_1_1_2_7_1_1_1_1_1_17"/>
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E7:2_1_1_2_7_1_1_1_1_1_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A202</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E91">A198</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:2_1_1_2_7_1_1_1_1_1_17"/><span class="lab"><font color="Green" title="E96">A203</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<a NAME="E9:2_1_1_2_7_1_1_1_1_1_17"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2
 
;<br/>
<a NAME="E10:2_1_1_2_7_1_1_1_1_1_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A204</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E91">A198</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E96">A203</font></span></a>, <a class="ref" href="xreal_1.html#T20" onmouseover="rs('xreal_1/T20')" onmouseout="rh()">XREAL_1:20</a></span>;<br/>
<a NAME="E11:2_1_1_2_7_1_1_1_1_1_17"/><span class="lab"><font color="Green" title="E98">A205</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_17_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_1_17_1"/>
( <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_17_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_17_1_1"/><span class="lab"><font color="Green" title="E99">A206</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_17_1_1"/><span class="lab"><font color="Green" title="E100">A207</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_17_1_1"/><span class="lab"><font color="Green" title="E101">A208</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E90">A197</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E99">A206</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A209</font></span>: 
<font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_17_1_1"/>
<font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E90">A197</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E99">A206</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A210</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c24">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E101">A208</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_17_1_1"/>
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E90">A197</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E99">A206</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_17_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E101">A208</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E100">A207</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_1_1_1_1_17_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_1_2"><span class="lab"><font color="Green" title="E33">A41</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E90">A197</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E99">A206</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E103">A210</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E102">A209</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_17_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_17_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_17_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_17_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A211</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_17_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_17_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/><span class="lab"><font color="Green" title="E100">A212</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><a NAME="E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E90">A197</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A212</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A213</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">assume </span><a NAME="E3:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E4:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E101">A213</font></span></a>, <a class="ref" href="topreal1.html#T23" onmouseover="rs('topreal1/T23')" onmouseout="rh()">TOPREAL1:23</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A214</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="goboard2.html#T4" onmouseover="rs('goboard2/T4')" onmouseout="rh()">GOBOARD2:4</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/>
( <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>) &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E90">A197</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E91">A198</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E95">A202</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E98">A205</font></span></a>, <a class="ref" href="topreal1.html#T21" onmouseover="rs('topreal1/T21')" onmouseout="rh()">TOPREAL1:21</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A215</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/><span class="lab"><font color="Green" title="E103">A216</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E92">A199</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A212</font></span></a>, <a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E96">A203</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A217</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A152</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E85">A153</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E13:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/><span class="lab"><font color="Green" title="E105">A218</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u1</font>
 <span class="kw">and </span><br/><a NAME="E14:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/>
<font color="Maroon" title="c25">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E15:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E16:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/>
<font color="Maroon" title="c25">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/><a NAME="E17:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/>
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T68" onmouseover="rs('rltopsp1/T68')" onmouseout="rh()">RLTOPSP1:68</a></span>;<br/><a NAME="E18:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A219</font></span>: 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<font color="Maroon" title="c25">u1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E87">A172</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E85">A153</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E104">A217</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A218</font></span></a>, <a class="ref" href="topreal1.html#T6" onmouseover="rs('topreal1/T6')" onmouseout="rh()">TOPREAL1:6</a></span>;<br/><a NAME="E19:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E20:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A220</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A212</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A221</font></span>: <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A212</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/><a NAME="E22:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/>
<font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E90">A197</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E94">A201</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A212</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E23:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A222</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c24">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E107">A220</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E24:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/>
<font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E93">A200</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E96">A203</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A212</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E103">A216</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><a NAME="E25:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A223</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<font color="Maroon" title="c25">u1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E91">A198</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E98">A205</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E108">A221</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A218</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E26:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c24">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="ref" href="topreal3.html#T19" onmouseover="rs('topreal3/T19')" onmouseout="rh()">TOPREAL3:19</a></span>;<br/><a NAME="E27:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A214</font></span></a>, <a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E109">A222</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E108">A221</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E106">A219</font></span></a>, <a class="txt" href="#E25:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E110">A223</font></span></a>, <a class="ref" href="xboole_1.html#T27" onmouseover="rs('xboole_1/T27')" onmouseout="rh()">XBOOLE_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E28:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A215</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_17_1_2"><span class="lab"><font color="Green" title="E99">A211</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A224</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">n1</font> = <font color="Maroon" title="c24">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">n</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E101">A225</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E95">A202</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E96">A203</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A226</font></span>: 
<font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A225</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E103">A227</font></span>: 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E91">A198</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E96">A203</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A225</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A228</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A152</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E85">A153</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E105">A229</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u1</font>
 <span class="kw">and </span><br/><a NAME="E11:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E106">A230</font></span>: 
<font color="Maroon" title="c26">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E12:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E13:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E107">A231</font></span>: 
<font color="Maroon" title="c26">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/><a NAME="E14:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E108">A232</font></span>: 
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E103">A227</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E15:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/><a NAME="E16:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A152</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E85">A153</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">u2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E18:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E109">A233</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">u2</font>
 <span class="kw">and </span><br/><a NAME="E19:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E110">A234</font></span>: 
<font color="Maroon" title="c27">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E20:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E21:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E111">A235</font></span>: 
<font color="Maroon" title="c27">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/><span class="kw">set </span><font color="Maroon" title="c28">lg</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/><a NAME="E22:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c27">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A107</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E104">A228</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E108">A232</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A229</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E109">A233</font></span></a></span>;<br/><a NAME="E23:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A236</font></span>: 
<font color="Maroon" title="c26">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E106">A230</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E107">A231</font></span></a></span>;<br/><a NAME="E24:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c27">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><a NAME="E25:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A237</font></span>: 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A83</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E109">A233</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E110">A234</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E111">A235</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>;<br/><a NAME="E26:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E27:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A225</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/><a NAME="E29:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A238</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<font color="Maroon" title="c26">u1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E90">A197</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E95">A202</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A226</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A229</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E30:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E91">A198</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E96">A203</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A225</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E31:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A225</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><a NAME="E32:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c26">u1</font>,<font color="Maroon" title="c27">u2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E91">A198</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E98">A205</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A226</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A229</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E109">A233</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E33:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A226</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A229</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E109">A233</font></span></a>, <a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E112">A236</font></span></a>, <a class="txt" href="#E29:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E114">A238</font></span></a>, <a class="txt" href="#E25:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E113">A237</font></span></a>, <a class="ref" href="topreal1.html#T8" onmouseover="rs('topreal1/T8')" onmouseout="rh()">TOPREAL1:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A239</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2"><span class="lab"><font color="Green" title="E100">A224</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A240</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c25">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">n</font>
 ;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A241</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c25">n1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E95">A202</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E101">A239</font></span></a>, <a class="ref" href="goboard2.html#T5" onmouseover="rs('goboard2/T5')" onmouseout="rh()">GOBOARD2:5</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E104">A242</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c25">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E95">A202</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E96">A203</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A243</font></span>: 
<font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E104">A242</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E94">A201</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E101">A239</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c25">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E92">A199</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E104">A242</font></span></a>, <a class="ref" href="goboard2.html#T5" onmouseover="rs('goboard2/T5')" onmouseout="rh()">GOBOARD2:5</a></span>;<br/><span class="kw">hence </span><a NAME="E13:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E89">A180</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_17"><span class="lab"><font color="Green" title="E97">A204</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E103">A241</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E105">A243</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E102">A240</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_1_1_1_1_17_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_1_1_1_1_17_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E46:2_1_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E90">A244</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_18"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font> or <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c25">ls</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> ) </span> } </span> ;<br/>
<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_1_18"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_18"/><span class="lab"><font color="Green" title="E91">A245</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">X</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_18"/><span class="lab"><font color="Green" title="E92">A246</font></span>: 
<font color="Maroon" title="c26">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_18"/><span class="lab"><font color="Green" title="E93">A247</font></span>: 
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c27">m</font>)
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_18"/><span class="lab"><font color="Green" title="E94">A248</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">m</font> &amp; <font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_18"><span class="lab"><font color="Green" title="E92">A246</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c28">q1</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">m</font>, <font color="Maroon" title="c29">q2</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E9:2_1_1_2_7_1_1_1_1_1_18"/><span class="lab"><font color="Green" title="E95">A249</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c27">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c28">q1</font>,<font color="Maroon" title="c29">q2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_18"><span class="lab"><font color="Green" title="E94">A248</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E10:2_1_1_2_7_1_1_1_1_1_18"/>
<font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_18"><span class="lab"><font color="Green" title="E94">A248</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E11:2_1_1_2_7_1_1_1_1_1_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A250</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_1_1_1_1_18"/>
<font color="Maroon" title="c27">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_18"><span class="lab"><font color="Green" title="E94">A248</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_1_1_1_1_18"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E14:2_1_1_2_7_1_1_1_1_1_18"/><span class="kw">then </span>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c28">q1</font>,<font color="Maroon" title="c29">q2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E87">A172</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E85">A153</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_18"><span class="lab"><font color="Green" title="E96">A250</font></span></a>, <a class="ref" href="topreal1.html#T6" onmouseover="rs('topreal1/T6')" onmouseout="rh()">TOPREAL1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:2_1_1_2_7_1_1_1_1_1_18"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E87">A172</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_18"><span class="lab"><font color="Green" title="E91">A245</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_18"><span class="lab"><font color="Green" title="E93">A247</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_1_18"><span class="lab"><font color="Green" title="E95">A249</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E47:2_1_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E91">A251</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_19"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_19_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_1_19_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_19_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_19_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_1_2_7_1_1_1_1_1_19_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2"><span class="lab"><font color="Green" title="E43">A52</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_19_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_19_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_19_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_19_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="ref" href="goboard2.html#T4" onmouseover="rs('goboard2/T4')" onmouseout="rh()">GOBOARD2:4</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_19_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A252</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E90">A244</font></span></a>, <a class="ref" href="xboole_1.html#T26" onmouseover="rs('xboole_1/T26')" onmouseout="rh()">XBOOLE_1:26</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_19_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c24">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span>;<br/><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_1_19_1_2_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_19_1_2_1"/><span class="kw">then </span>
(  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_19_1_2"><span class="lab"><font color="Green" title="E92">A252</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_1_1_1_1_19_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E80">A134</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_1_1_1_1_19_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E48:2_1_1_2_7_1_1_1_1_1"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> &amp; <font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> holds <br/> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Olive" title="b1">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Olive" title="b2">m</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E92">A253</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E2:2_1_1_2_7_1_1_1_1_1_20"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E93">A254</font></span>: <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> = 
<font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">qq</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">l1</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c26">l2</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> ) </span> } </span> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c27">n</font>, <font color="Maroon" title="c28">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c28">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c27">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> &amp; <font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> &amp; <font color="Maroon" title="c28">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> &amp; <font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> implies  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E94">A255</font></span>: 
<font color="Maroon" title="c28">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E95">A256</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E96">A257</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E97">A258</font></span>: 
<font color="Maroon" title="c28">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">and </span><br/><a NAME="E9:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E98">A259</font></span>: 
<font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/>

<a NAME="E10:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E99">A260</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E95">A256</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E11:2_1_1_2_7_1_1_1_1_1_20"/>
<font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A77</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_1_1_1_1_20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A261</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_1_1_1_1_20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A262</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E14:2_1_1_2_7_1_1_1_1_1_20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A263</font></span>: 
( <font color="Maroon" title="c24">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c24">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a></span>;<br/>
<a NAME="E15:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E103">A264</font></span>: 
<font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">qq</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E100">A261</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/>
<a NAME="E16:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E104">A265</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E101">A262</font></span></a></span>;<br/>
<a NAME="E17:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E105">A266</font></span>: 
<font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E98">A259</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E18:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E106">A267</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E98">A259</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E19:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E107">A268</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E96">A257</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E20:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E108">A269</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E96">A257</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E21:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E109">A270</font></span>: 
<font color="Maroon" title="c24">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<a NAME="E22:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E110">A271</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E97">A258</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c29">ql</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c24">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/>
<a NAME="E23:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E111">A272</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E24:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E112">A273</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<a NAME="E25:2_1_1_2_7_1_1_1_1_1_20"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E100">A261</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E113">A274</font></span>:  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c24">qq</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E92">A253</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E93">A254</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E103">A264</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c24">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A83</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E102">A263</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E109">A270</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>
;<br/>
<a NAME="E27:2_1_1_2_7_1_1_1_1_1_20"/><span class="lab"><font color="Green" title="E114">A275</font></span>: 
<font color="Maroon" title="c28">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E28:2_1_1_2_7_1_1_1_1_1_20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A276</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E94">A255</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_1_20_3_1"/>
( <font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_20_3_1_1"/><span class="lab"><font color="Green" title="E116">A277</font></span>: 
<font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_20_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c28">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E114">A275</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_20_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A278</font></span>: 
<font color="Maroon" title="c28">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E110">A271</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_20_3_1_1"/>
<font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E106">A267</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_20_3_1_1"><span class="lab"><font color="Green" title="E116">A277</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_20_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A279</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c28">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_20_3_1_1"><span class="lab"><font color="Green" title="E117">A278</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_20_3_1_1"/><span class="lab"><font color="Green" title="E119">A280</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E115">A276</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_20_3_1_1"><span class="lab"><font color="Green" title="E116">A277</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_20_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E111">A272</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_20_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A281</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E99">A260</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_1_20_3_1_1"/>
<font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E107">A268</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_20_3_1_1"><span class="lab"><font color="Green" title="E119">A280</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1_20_3_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c27">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_20_3_1_1"><span class="lab"><font color="Green" title="E120">A281</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><span class="kw">hence </span><a NAME="E11:2_1_1_2_7_1_1_1_1_1_20_3_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:2_1_1_2"><span class="lab"><font color="Green" title="E34">A42</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E94">A255</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_20_3_1_1"><span class="lab"><font color="Green" title="E118">A279</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_20_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A282</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">m1</font> = <font color="Maroon" title="c28">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1"/>
( <font color="Maroon" title="c28">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or <font color="Maroon" title="c28">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E117">A283</font></span>: 
<font color="Maroon" title="c28">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E118">A284</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c31">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>) or <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_1"/>
<font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c32">px</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E119">A285</font></span>: 
( <font color="Maroon" title="c32">px</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Maroon" title="c32">px</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E120">A286</font></span>: 
<font color="Maroon" title="c24">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">px</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E121">A287</font></span>: 
<font color="Maroon" title="c32">px</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E113">A274</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E117">A283</font></span></a></span>;<br/>
<a NAME="E6:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">px</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E104">A265</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E120">A286</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_1"/>
<font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A152</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E119">A285</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E121">A287</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"/>
<font color="Maroon" title="c27">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E94">A255</font></span></a>, <a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E111">A272</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E117">A283</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A288</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E99">A260</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><a NAME="E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E94">A255</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E107">A268</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E117">A283</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A289</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">assume </span><a NAME="E3:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E120">A289</font></span></a>, <a class="ref" href="topreal1.html#T23" onmouseover="rs('topreal1/T23')" onmouseout="rh()">TOPREAL1:23</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A290</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="goboard2.html#T4" onmouseover="rs('goboard2/T4')" onmouseout="rh()">GOBOARD2:4</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E121">A291</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E94">A255</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E107">A268</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E117">A283</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A292</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c27">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E119">A288</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E94">A255</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E99">A260</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E117">A283</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A293</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E120">A290</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E118">A284</font></span></a>, <a class="ref" href="xboole_1.html#T27" onmouseover="rs('xboole_1/T27')" onmouseout="rh()">XBOOLE_1:27</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c31">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span>;<br/><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_3"/><span class="lab"><font color="Green" title="E124">A294</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A295</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_3"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E123">A293</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E124">A294</font></span></a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A296</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_3"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>;<br/><span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:2_1_1_2"><span class="lab"><font color="Green" title="E32">A40</font></span></a>, <a class="txt" href="#E30:2_1_1_2"><span class="lab"><font color="Green" title="E33">A41</font></span></a>, <a class="txt" href="#E31:2_1_1_2"><span class="lab"><font color="Green" title="E34">A42</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E94">A255</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E117">A283</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E119">A288</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E121">A291</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E122">A292</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E125">A295</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E126">A296</font></span></a>, <a class="ref" href="goboard2.html#T2" onmouseover="rs('goboard2/T2')" onmouseout="rh()">GOBOARD2:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E14:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"/>
<font color="Maroon" title="c28">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A297</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_20_3_1_2"><span class="lab"><font color="Green" title="E116">A282</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A298</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"/>
<font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A299</font></span>: 
<font color="Maroon" title="c30">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E105">A266</font></span></a>, <a class="txt" href="#E24:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E112">A273</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"/>
<font color="Maroon" title="c28">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A300</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c30">m1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E105">A266</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E117">A297</font></span></a>, <a class="ref" href="goboard2.html#T5" onmouseover="rs('goboard2/T5')" onmouseout="rh()">GOBOARD2:5</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A298</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E119">A299</font></span></a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A301</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E122">A302</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E111">A272</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A303</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E99">A260</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E107">A268</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E122">A302</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c27">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E123">A303</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E99">A260</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E122">A302</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E91">A251</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E121">A301</font></span></a>, <a class="ref" href="xboole_1.html#T3" onmouseover="rs('xboole_1/T3')" onmouseout="rh()">XBOOLE_1:3</a>, <a class="ref" href="xboole_1.html#T27" onmouseover="rs('xboole_1/T27')" onmouseout="rh()">XBOOLE_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A304</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">n1</font> = <font color="Maroon" title="c27">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E123">A305</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E124">A306</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">n</font> or <font color="Maroon" title="c27">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E125">A307</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c32">q1</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c30">m1</font>, <font color="Maroon" title="c33">q2</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c34">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c35">q1l</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">v</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E126">A308</font></span>: 
( <font color="Maroon" title="c32">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c32">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c33">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c33">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><span class="kw">assume </span><a NAME="E3:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E127">A309</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A310</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E127">A309</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A311</font></span>: 
 ex <font color="Olive" title="b1">qx</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">qx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> &amp; <font color="Olive" title="b1">qx</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c24">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qx</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">qx</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E113">A274</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E125">A307</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E130">A312</font></span>: 
<font color="Maroon" title="c30">m1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A298</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E119">A299</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A313</font></span>: 
<font color="Maroon" title="c32">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E132">A314</font></span>: 
<font color="Maroon" title="c30">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A298</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E119">A299</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A315</font></span>: 
<font color="Maroon" title="c33">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c30">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A316</font></span>: 
<font color="Maroon" title="c33">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A107</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E130">A312</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E132">A314</font></span></a></span>;<br/> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c30">m1</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c33">q2</font>,<font color="Maroon" title="c32">q1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A298</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E119">A299</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">v</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E131">A313</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E133">A315</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E134">A316</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E126">A308</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>
;<br/><a NAME="E14:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A317</font></span>: 
 ex <font color="Olive" title="b1">qy</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">qy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> &amp; <font color="Olive" title="b1">qy</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qy</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">qy</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A300</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E128">A310</font></span></a></span>;<br/><a NAME="E15:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c30">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c31">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c31">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E94">A255</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E123">A305</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E16:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c30">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E17:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c32">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E78">A107</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E101">A262</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E130">A312</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E18:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E129">A311</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E135">A317</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c27">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E122">A304</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A318</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c31">n1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E108">A269</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E124">A306</font></span></a>, <a class="ref" href="goboard2.html#T5" onmouseover="rs('goboard2/T5')" onmouseout="rh()">GOBOARD2:5</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c30">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c31">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_20"><span class="lab"><font color="Green" title="E94">A255</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E123">A305</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E79">A133</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A300</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E125">A318</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E12:2_1_1_2_7_1_1_1_1_1_20_3_1_2_1_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_1_1_1_1_20_3_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E30:2_1_1_2_7_1_1_1_1_1_20"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E49:2_1_1_2_7_1_1_1_1_1"/>
<font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#T1" onmouseover="rs('goboard2/T1')" onmouseout="rh()">GOBOARD2:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_21"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c24">p</font> = <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c25">q</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1;<br/><a NAME="E1:2_1_1_2_7_1_1_1_1_1_21"/>
<font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A77</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_21"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_21"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_21"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_1_1_1_1_21"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E51:2_1_1_2_7_1_1_1_1_1"/>
<font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:2_1_1_2"><span class="lab"><font color="Green" title="E35">A43</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E77">A104</font></span></a>, <a class="ref" href="goboard2.html#T8" onmouseover="rs('goboard2/T8')" onmouseout="rh()">GOBOARD2:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )</span><br/><span class="kw">thus </span><a NAME="E52:2_1_1_2_7_1_1_1_1_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c24">lg</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">lf</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span> } </span> ;<br/>
<a NAME="E1:2_1_1_2_7_1_1_1_1_1_22"/><span class="lab"><font color="Green" title="E92">A319</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E93">A320</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> holds <br/> for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font> holds <br/>( <font color="Olive" title="b2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b2">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c26">j</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">j</font> &amp; <font color="Maroon" title="c26">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> implies  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">j</font> holds <br/>( <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_1"/><span class="lab"><font color="Green" title="E94">A321</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">j</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_22_1"/><span class="lab"><font color="Green" title="E95">A322</font></span>: 
<font color="Maroon" title="c26">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">j</font> holds <br/>( <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c27">w</font> = <font color="Maroon" title="c26">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_1"><span class="lab"><font color="Green" title="E94">A321</font></span></a>, <a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c28">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c28">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">j</font> implies ( <font color="Maroon" title="c28">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> ) )</span><br/><span class="kw">assume </span><a NAME="E4:2_1_1_2_7_1_1_1_1_1_22_1"/><span class="lab"><font color="Green" title="E96">A323</font></span>: 
<font color="Maroon" title="c28">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">j</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c28">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> )</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c28">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_1_1_1"/>
( <font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or <font color="Maroon" title="c26">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_1_1_1_1"/><span class="lab"><font color="Green" title="E97">A324</font></span>: 
<font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c28">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> )</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_22_1_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_22_1_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A325</font></span>: <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_1_1_1_1_22_1_1_1_1"/>
<font color="Maroon" title="c28">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_22_1"><span class="lab"><font color="Green" title="E96">A323</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_1_1_1_1"><span class="lab"><font color="Green" title="E97">A324</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> )</span><br/><span class="kw">thus </span><a NAME="E6:2_1_1_2_7_1_1_1_1_1_22_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E53:2_1_1_2"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A77</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E64">A79</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E66">A81</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_22_1"><span class="lab"><font color="Green" title="E96">A323</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_1_1_1_1"><span class="lab"><font color="Green" title="E97">A324</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_22_1_1_1_1"><span class="lab"><font color="Green" title="E98">A325</font></span></a>, <a class="ref" href="seqm_3.html#D1" onmouseover="rs('seqm_3/D1')" onmouseout="rh()">SEQM_3:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c28">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font></span><br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_22_1_1_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">l1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:2_1_1_2_7_1_1_1_1_1_22_1_1_1_1"/>
<font color="Maroon" title="c28">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E63">A78</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_22_1"><span class="lab"><font color="Green" title="E96">A323</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_1_1_1_1"><span class="lab"><font color="Green" title="E97">A324</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_22_1_1_1_1"><span class="lab"><font color="Green" title="E98">A325</font></span></a>, <a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_1_1_1_2"/>
<font color="Maroon" title="c26">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c28">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> )</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_22_1_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_1"><span class="lab"><font color="Green" title="E94">A321</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_22_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_22_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A326</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_22_1_1_1_2"/><span class="lab"><font color="Green" title="E98">A327</font></span>: 
<font color="Maroon" title="c27">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22"><span class="lab"><font color="Green" title="E92">A319</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_22_1"><span class="lab"><font color="Green" title="E95">A322</font></span></a>, <a class="ref" href="xreal_1.html#T20" onmouseover="rs('xreal_1/T20')" onmouseout="rh()">XREAL_1:20</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_22_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A328</font></span>: 
<font color="Maroon" title="c27">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_22_1_1_1_2"><span class="lab"><font color="Green" title="E97">A326</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_22_1_1_1_2"/>
<font color="Maroon" title="c27">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">j</font>
 ;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_22_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_22_1_1_1_2"><span class="lab"><font color="Green" title="E97">A326</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_22_1_1_1_2"><span class="lab"><font color="Green" title="E98">A327</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_1_1_1_1_22_1_1_1_2"/>
( <font color="Maroon" title="c28">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E76">A95</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_22_1"><span class="lab"><font color="Green" title="E96">A323</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_22_1_1_1_2"><span class="lab"><font color="Green" title="E99">A328</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_1_1_1_1_22_1"/>
( <font color="Maroon" title="c28">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E3:2_1_1_2_7_1_1_1_1_1_22"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> or <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_22_2"/><span class="lab"><font color="Green" title="E94">A329</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">X</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_22_2"/><span class="lab"><font color="Green" title="E95">A330</font></span>: 
<font color="Maroon" title="c27">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c28">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_22_2"/><span class="lab"><font color="Green" title="E96">A331</font></span>: 
<font color="Maroon" title="c27">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">i</font>)
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_22_2"/><span class="lab"><font color="Green" title="E97">A332</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">i</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_22_2"/><span class="lab"><font color="Green" title="E98">A333</font></span>: 
<font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_22_2"><span class="lab"><font color="Green" title="E95">A330</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_2_1"/>
( <font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or <font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_2_1_1"/><span class="lab"><font color="Green" title="E99">A334</font></span>: 
<font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_22_2_1_1"/>
<font color="Maroon" title="c28">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_22_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c28">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_2_1_1"><span class="lab"><font color="Green" title="E99">A334</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_22_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A335</font></span>: 
<font color="Maroon" title="c28">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_22_2"><span class="lab"><font color="Green" title="E97">A332</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_22_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_22_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_2_1_1"><span class="lab"><font color="Green" title="E99">A334</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_22_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c28">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_22_2"><span class="lab"><font color="Green" title="E96">A331</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_22_2_1_1"><span class="lab"><font color="Green" title="E100">A335</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_22_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_22_2"><span class="lab"><font color="Green" title="E97">A332</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_2_1_1"><span class="lab"><font color="Green" title="E99">A334</font></span></a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_1_22_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A336</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_22_2"><span class="lab"><font color="Green" title="E94">A329</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1_22_2_1_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T20" onmouseover="rs('topreal3/T20')" onmouseout="rh()">TOPREAL3:20</a></span>;<br/><span class="kw">hence </span><a NAME="E11:2_1_1_2_7_1_1_1_1_1_22_2_1_1"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_1_22_2_1_1"><span class="lab"><font color="Green" title="E101">A336</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_2_1_2"/><span class="lab"><font color="Green" title="E99">A337</font></span>: 
<font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c29">q1</font> = <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">i</font>, <font color="Maroon" title="c30">q2</font> = <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_22_2_1_2"/><span class="lab"><font color="Green" title="E100">A338</font></span>: 
<font color="Maroon" title="c28">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_22_2"><span class="lab"><font color="Green" title="E98">A333</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_22_2_1_2"/><span class="lab"><font color="Green" title="E101">A339</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E99">A337</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_22_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A340</font></span>: 
<font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_22"><span class="lab"><font color="Green" title="E93">A320</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E100">A338</font></span></a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_22_2_1_2"/><span class="lab"><font color="Green" title="E103">A341</font></span>: 
<font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_22"><span class="lab"><font color="Green" title="E93">A320</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E101">A339</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E100">A338</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_22_2_1_2"/><span class="lab"><font color="Green" title="E104">A342</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_22"><span class="lab"><font color="Green" title="E93">A320</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E101">A339</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E100">A338</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_22_2_1_2"/>
<font color="Maroon" title="c30">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_22"><span class="lab"><font color="Green" title="E93">A320</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_22_2"><span class="lab"><font color="Green" title="E98">A333</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E99">A337</font></span></a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_1_22_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A343</font></span>: 
<font color="Maroon" title="c30">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E102">A340</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1_22_2_1_2"/><span class="lab"><font color="Green" title="E106">A344</font></span>: 
<font color="Maroon" title="c30">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_22"><span class="lab"><font color="Green" title="E93">A320</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_22_2"><span class="lab"><font color="Green" title="E98">A333</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E99">A337</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_1_22_2_1_2"/><span class="lab"><font color="Green" title="E107">A345</font></span>: 
( <font color="Maroon" title="c29">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp;  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c30">q2</font>,<font color="Maroon" title="c29">q1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_22_2"><span class="lab"><font color="Green" title="E97">A332</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_22_2"><span class="lab"><font color="Green" title="E98">A333</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_1_1_1_1_22_2_1_2"/><span class="lab"><font color="Green" title="E108">A346</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_22"><span class="lab"><font color="Green" title="E93">A320</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_22_2"><span class="lab"><font color="Green" title="E98">A333</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E99">A337</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_2_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1"/>
( <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  or <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  or <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"/>
<font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E105">A343</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E107">A345</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E109">A347</font></span>: 
( <font color="Maroon" title="c31">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">x</font> &amp; <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E110">A348</font></span>: 
( <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_22_2"><span class="lab"><font color="Green" title="E94">A329</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_22_2"><span class="lab"><font color="Green" title="E96">A331</font></span></a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E103">A341</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E108">A346</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E110">A348</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A349</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A152</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E102">A340</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E109">A347</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E111">A349</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_2"/>
<font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c29">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E105">A343</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E107">A345</font></span></a>, <a class="ref" href="rltopsp1.html#T70" onmouseover="rs('rltopsp1/T70')" onmouseout="rh()">RLTOPSP1:70</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_22_2"><span class="lab"><font color="Green" title="E94">A329</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_22_2"><span class="lab"><font color="Green" title="E96">A331</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A350</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A152</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E102">A340</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E104">A342</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E103">A341</font></span></a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E109">A350</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"/>
<font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p1</font> where <font color="Olive" title="b1">p1</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E105">A343</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E107">A345</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"/><span class="lab"><font color="Green" title="E109">A351</font></span>: 
( <font color="Maroon" title="c31">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">x</font> &amp; <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"/><span class="lab"><font color="Green" title="E110">A352</font></span>: 
( <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_22_2"><span class="lab"><font color="Green" title="E94">A329</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_22_2"><span class="lab"><font color="Green" title="E96">A331</font></span></a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E104">A342</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E106">A344</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E110">A352</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A353</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A152</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E102">A340</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E109">A351</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_22_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E111">A353</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E14:2_1_1_2_7_1_1_1_1_1_22_2_1_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c26">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> or <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:2_1_1_2_7_1_1_1_1_1_22"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font></span><br/>

<a NAME="E5:2_1_1_2_7_1_1_1_1_1_22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A354</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E1:2_1_1_2"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="ref" href="goboard2.html#T3" onmouseover="rs('goboard2/T3')" onmouseout="rh()">GOBOARD2:3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_3"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_3"/>
( <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> or <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_22"><span class="lab"><font color="Green" title="E94">A354</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_3_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_3_1"/><span class="lab"><font color="Green" title="E95">A355</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_22_3_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#T6" onmouseover="rs('goboard2/T6')" onmouseout="rh()">GOBOARD2:6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_1_1_1_1_22_3_1"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_3_1"><span class="lab"><font color="Green" title="E95">A355</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_3_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_3_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">p1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_22_3_2"/><span class="lab"><font color="Green" title="E95">A356</font></span>: 
<font color="Maroon" title="c27">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_22_3_2"/><span class="lab"><font color="Green" title="E96">A357</font></span>: 
<font color="Maroon" title="c27">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_22_3_2"/><span class="lab"><font color="Green" title="E97">A358</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_22_3_2"/><span class="lab"><font color="Green" title="E98">A359</font></span>: 
<font color="Maroon" title="c27">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A152</font></span></a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> &amp; (  for <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c27">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) );<br/><div><span class="lab"><font color="Green" title="E99">A360</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_3_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]</span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c28">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><span class="kw">take </span><font color="Maroon" title="c29">n</font> = <font color="Maroon" title="c28">n</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c29">n</font>]</span><br/><span class="kw">thus </span><a NAME="E2:2_1_1_2_7_1_1_1_1_1_22_3_2_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c29">n</font>]
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_3_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_3_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">n</font> &amp; <font color="Maroon" title="c29">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22"><span class="lab"><font color="Green" title="E92">A319</font></span></a>, <a class="ref" href="xreal_1.html#T31" onmouseover="rs('xreal_1/T31')" onmouseout="rh()">XREAL_1:31</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">n</font> holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c27">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/>

<a NAME="E2:2_1_1_2_7_1_1_1_1_1_22_3_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E3:2_1_1_2_7_1_1_1_1_1_22_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A361</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c30">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c30">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">n</font> implies <font color="Maroon" title="c30">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c27">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E4:2_1_1_2_7_1_1_1_1_1_22_3_2_1_1"/>
<font color="Maroon" title="c30">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c27">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/>

<span class="kw">then </span><font color="Maroon" title="c30">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_22_3_2_1_1"><span class="lab"><font color="Green" title="E100">A361</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_1_1_1_1_22_3_2_1_1"/>
<font color="Maroon" title="c30">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c27">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E98">A359</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E8:2_1_1_2_7_1_1_1_1_1_22_3_2"/><span class="lab"><font color="Green" title="E100">A362</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c28">ma</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1_22_3_2"/><span class="lab"><font color="Green" title="E101">A363</font></span>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c28">ma</font>] &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S6" onmouseover="rs('nat_1/S6')" onmouseout="rh()">NAT_1:sch 6</a>(<a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E100">A362</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E99">A360</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c29">ma</font> = <font color="Maroon" title="c28">ma</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_3_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_3_2_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1"/>
( <font color="Maroon" title="c29">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> or <font color="Maroon" title="c29">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><span class="lab"><font color="Green" title="E102">A364</font></span>: 
<font color="Maroon" title="c29">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/>
<font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A77</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A365</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22"><span class="lab"><font color="Green" title="E92">A319</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E102">A364</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A366</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">m1</font> = <font color="Maroon" title="c29">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c31">q</font> = <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c30">m1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><span class="lab"><font color="Green" title="E105">A367</font></span>: 
<font color="Maroon" title="c29">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E102">A364</font></span></a>, <a class="ref" href="xreal_1.html#T43" onmouseover="rs('xreal_1/T43')" onmouseout="rh()">XREAL_1:43</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A368</font></span>: 
<font color="Maroon" title="c31">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_22"><span class="lab"><font color="Green" title="E93">A320</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E104">A366</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><span class="lab"><font color="Green" title="E107">A369</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_22"><span class="lab"><font color="Green" title="E93">A320</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E105">A367</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E104">A366</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c32">lq</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">e</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">e</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/><a NAME="E12:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><span class="lab"><font color="Green" title="E108">A370</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font>
 ;<br/><a NAME="E13:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><span class="lab"><font color="Green" title="E109">A371</font></span>: 
<font color="Maroon" title="c19">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E103">A365</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A372</font></span>: <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">ma</font> = 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22"><span class="lab"><font color="Green" title="E92">A319</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E102">A364</font></span></a>, <a class="ref" href="finseq_4.html#T69" onmouseover="rs('finseq_4/T69')" onmouseout="rh()">FINSEQ_4:69</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E109">A371</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E108">A370</font></span></a></span>
;<br/><a NAME="E15:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E101">A363</font></span></a></span>;<br/><a NAME="E16:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A373</font></span>: 
<font color="Maroon" title="c27">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E97">A358</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E17:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E18:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A374</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E104">A366</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E19:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><span class="lab"><font color="Green" title="E113">A375</font></span>: 
<font color="Maroon" title="c30">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">ma</font>
 ;<br/><a NAME="E20:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c31">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp;  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c30">m1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span>,<font color="Maroon" title="c31">q</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E102">A364</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E110">A372</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E112">A374</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E21:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c30">m1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">e</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">e</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E81">A145</font></span></a>, <a class="txt" href="#E32:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E83">A151</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E106">A368</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E107">A369</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>;<br/><a NAME="E22:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A376</font></span>: 
<font color="Maroon" title="c27">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c30">m1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E96">A357</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E111">A373</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E107">A369</font></span></a></span>;<br/><a NAME="E23:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c30">m1</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E102">A364</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E112">A374</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E113">A375</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E24:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E95">A356</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E114">A376</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/>
<font color="Maroon" title="c29">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c29">ma</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E101">A363</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A377</font></span>: 
<font color="Maroon" title="c29">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c30">qa</font> = <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">ma</font>, <font color="Maroon" title="c31">qa1</font> = <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c32">lma</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c31">qa1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/><span class="lab"><font color="Green" title="E103">A378</font></span>: 
<font color="Maroon" title="c31">qa1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">qa1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">qa1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/><span class="lab"><font color="Green" title="E104">A379</font></span>: 
<font color="Maroon" title="c27">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E101">A363</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/><span class="lab"><font color="Green" title="E105">A380</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E101">A363</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A381</font></span>: 
<font color="Maroon" title="c31">qa1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_22"><span class="lab"><font color="Green" title="E93">A320</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E102">A377</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E107">A382</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c27">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">qa1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2_1"/>
<font color="Maroon" title="c27">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">qa1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c27">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c29">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E101">A363</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E102">A377</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E105">A380</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E10:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/><span class="lab"><font color="Green" title="E108">A383</font></span>: 
( <font color="Maroon" title="c30">qa</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c30">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">qa</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">qa</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_1_22"><span class="lab"><font color="Green" title="E93">A320</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E101">A363</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/><span class="lab"><font color="Green" title="E109">A384</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E101">A363</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c29">ma</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c31">qa1</font>,<font color="Maroon" title="c30">qa</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E102">A377</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c31">qa1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E104">A379</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E107">A382</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E106">A381</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E108">A383</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E103">A378</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
;<br/><a NAME="E13:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A385</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c29">ma</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E95">A356</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E96">A357</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E104">A379</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E107">A382</font></span></a></span>;<br/><a NAME="E14:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c29">ma</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E102">A377</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E109">A384</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E110">A385</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:2_1_1_2_7_1_1_1_1_1_22_3_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E53:2_1_1_2_7_1_1_1_1_1"/><span class="lab"><font color="Green" title="E92">A386</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/><a NAME="E54:2_1_1_2_7_1_1_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E55:2_1_1_2_7_1_1_1_1_1"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2"><span class="lab"><font color="Green" title="E37">A45</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E16:2_1_1_2"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )</span><br/><a NAME="E57:2_1_1_2_7_1_1_1_1_1"/>
<font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A77</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E58:2_1_1_2_7_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A387</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E59:2_1_1_2_7_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A388</font></span>: 
<font color="Maroon" title="c19">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> = 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E53:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E92">A386</font></span></a>, <a class="ref" href="finseq_4.html#T69" onmouseover="rs('finseq_4/T69')" onmouseout="rh()">FINSEQ_4:69</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c23">m1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A90</font></span></a>, <a class="txt" href="#E59:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E94">A388</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E12:2_1_1_2"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1"><span class="lab"><font color="Green" title="E61">A76</font></span></a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span><br/><span class="kw">thus </span><a NAME="E61:2_1_1_2_7_1_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A89</font></span></a>, <a class="txt" href="#E58:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E93">A387</font></span></a>, <a class="txt" href="#E53:2_1_1_2_7_1_1_1_1_1"><span class="lab"><font color="Green" title="E92">A386</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_2"/><span class="lab"><font color="Green" title="E62">A389</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E9:2_1_1_2"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E12:2_1_1_2"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_2"><span class="lab"><font color="Green" title="E62">A389</font></span></a>, <a class="ref" href="partfun2.html#T10" onmouseover="rs('partfun2/T10')" onmouseout="rh()">PARTFUN2:10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3"><span class="kw">case </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3"/><span class="lab"><font color="Green" title="E62">A390</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> )</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3"/>
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E55:2_1_1_2"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A391</font></span>: 
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A392</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3"/>
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E55:2_1_1_2"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A393</font></span>: 
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A394</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A395</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E53:2_1_1_2"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E62">A390</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E64">A392</font></span></a>, <a class="ref" href="seqm_3.html#D1" onmouseover="rs('seqm_3/D1')" onmouseout="rh()">SEQM_3:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">l</font> = <font color="Maroon" title="c7">j2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">i2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E62">A390</font></span></a>, <a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1</span>)</span>);<br/><span class="kw">consider </span><font color="Maroon" title="c20">g2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E11:2_1_1_2_7_1_1_1_1_3"/><span class="lab"><font color="Green" title="E68">A396</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> holds <br/><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_4.html#S2" onmouseover="rs('finseq_4/S2')" onmouseout="rh()">FINSEQ_4:sch 2</a>();<br/></span><span class="kw">take </span><font color="Maroon" title="c21">g</font> = <font color="Maroon" title="c16">g1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c20">g2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )</span><br/><a NAME="E12:2_1_1_2_7_1_1_1_1_3"/><span class="lab"><font color="Green" title="E69">A397</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><div><span class="lab"><font color="Green" title="E70">A398</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> holds <br/>( <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> implies ( <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> ) )</span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_3_1"/><span class="lab"><font color="Green" title="E71">A399</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><span class="kw">assume </span><a NAME="E2:2_1_1_2_7_1_1_1_1_3_1"/><span class="lab"><font color="Green" title="E72">A400</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )</span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A401</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_1"/>
<font color="Maroon" title="c7">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A402</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E73">A401</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E72">A400</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E71">A399</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_1_1_1_3_1"/>
<font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E74">A402</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font></span><br/><span class="kw">hence </span><a NAME="E10:2_1_1_2_7_1_1_1_1_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_1_2"><span class="lab"><font color="Green" title="E20">A22</font></span></a>, <a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> holds <br/> ex <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b3">k</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font>) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> implies  ex <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b2">k</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k</font>) ) )</span><br/><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_3_2"/><span class="lab"><font color="Green" title="E71">A403</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b2">k</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k</font>) )</span><br/><span class="kw">take </span><font color="Maroon" title="c23">m</font> = <font color="Maroon" title="c8">i1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">m</font>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c23">m</font>,<font color="Olive" title="b1">k</font>) )</span><br/><span class="kw">take </span><font color="Maroon" title="c24">k</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c24">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c24">k</font>) )</span><br/><span class="kw">thus </span><a NAME="E2:2_1_1_2_7_1_1_1_1_3_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c24">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c24">k</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E70">A398</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E69">A397</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_2"><span class="lab"><font color="Green" title="E71">A403</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E15:2_1_1_2_7_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A404</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:2_1_1_2"><span class="lab"><font color="Green" title="E60">A75</font></span></a>, <a class="ref" href="goboard1.html#T23" onmouseover="rs('goboard1/T23')" onmouseout="rh()">GOBOARD1:23</a></span>;<br/><a NAME="E16:2_1_1_2_7_1_1_1_1_3"/><span class="lab"><font color="Green" title="E72">A405</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E44:2_1_1_2"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E63">A391</font></span></a>, <a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><div><span class="lab"><font color="Green" title="E73">A406</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> holds <br/>( <font color="Olive" title="b2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b2">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> &amp; <font color="Olive" title="b2">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> holds <br/>( <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c23">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p</font> implies ( <font color="Maroon" title="c23">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c23">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c23">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> &amp; <font color="Maroon" title="c23">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_3_3"/><span class="lab"><font color="Green" title="E74">A407</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_3_3"/><span class="lab"><font color="Green" title="E75">A408</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c23">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c23">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> &amp; <font color="Maroon" title="c23">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_3"/><span class="lab"><font color="Green" title="E76">A409</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E74">A407</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c24">n1</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font>;<br/><span class="kw">set </span><font color="Maroon" title="c25">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>);<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_3"/><span class="lab"><font color="Green" title="E77">A410</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E70">A398</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E69">A397</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E74">A407</font></span></a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A411</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E49:2_1_1_2"><span class="lab"><font color="Green" title="E52">A67</font></span></a>, <a class="txt" href="#E44:2_1_1_2"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="ref" href="seqm_3.html#D10" onmouseover="rs('seqm_3/D10')" onmouseout="rh()">SEQM_3:def 10</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_3"/>
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:2_1_1_2"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E70">A398</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E69">A397</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E74">A407</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A412</font></span>: 
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E77">A410</font></span></a>, <a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A413</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E77">A410</font></span></a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_3_3"/>
<font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E69">A397</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E74">A407</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A414</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_3_3"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:2_1_1_2"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E77">A410</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E79">A412</font></span></a>, <a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E12:2_1_1_2_7_1_1_1_1_3_3"/>
( <font color="Maroon" title="c23">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c23">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E53:2_1_1_2"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E64">A392</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E66">A394</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E72">A405</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E75">A408</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E76">A409</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E77">A410</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E81">A414</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E78">A411</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E80">A413</font></span></a>, <a class="ref" href="seq_4.html#T137" onmouseover="rs('seq_4/T137')" onmouseout="rh()">SEQ_4:137</a>, <a class="ref" href="xreal_1.html#T31" onmouseover="rs('xreal_1/T31')" onmouseout="rh()">XREAL_1:31</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> &amp; <font color="Maroon" title="c23">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><a NAME="E13:2_1_1_2_7_1_1_1_1_3_3"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">l1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E14:2_1_1_2_7_1_1_1_1_3_3"/>
<font color="Maroon" title="c23">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E75">A408</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E76">A409</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E77">A410</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E79">A412</font></span></a>, <a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/><a NAME="E15:2_1_1_2_7_1_1_1_1_3_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E69">A397</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E74">A407</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E16:2_1_1_2_7_1_1_1_1_3_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><span class="kw">hence </span><a NAME="E17:2_1_1_2_7_1_1_1_1_3_3"/>
<font color="Maroon" title="c23">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E53:2_1_1_2"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E64">A392</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E75">A408</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E76">A409</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E77">A410</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_3"><span class="lab"><font color="Green" title="E80">A413</font></span></a>, <a class="ref" href="seqm_3.html#D1" onmouseover="rs('seqm_3/D1')" onmouseout="rh()">SEQM_3:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E18:2_1_1_2_7_1_1_1_1_3"/><span class="lab"><font color="Green" title="E74">A415</font></span>: 
<font color="Maroon" title="c20">g2</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topreal1.html#D5" onmouseover="rs('topreal1/D5')" onmouseout="rh()">TOPREAL1:def 5</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font> or  not <font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c23">p</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font>;<br/>
<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_3_4"/><span class="lab"><font color="Green" title="E75">A416</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font> &amp; <font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<a NAME="E2:2_1_1_2_7_1_1_1_1_3_4"/><span class="kw">then </span>
<font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E3:2_1_1_2_7_1_1_1_1_3_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A417</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E73">A406</font></span></a></span>;<br/>
<a NAME="E4:2_1_1_2_7_1_1_1_1_3_4"/>
<font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_4"><span class="lab"><font color="Green" title="E75">A416</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_1_1_1_3_4"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E73">A406</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_4"><span class="lab"><font color="Green" title="E76">A417</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">m</font> holds <br/> not <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">n</font>, <font color="Maroon" title="c23">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c22">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">m</font> implies  not <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">m</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_3_5"/><span class="lab"><font color="Green" title="E75">A418</font></span>: 
( <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_3_5"/><span class="lab"><font color="Green" title="E76">A419</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">m</font></span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_5"/><span class="lab"><font color="Green" title="E77">A420</font></span>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c22">n</font></span>)</span>) &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">m</font></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_5"><span class="lab"><font color="Green" title="E75">A418</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E4:2_1_1_2_7_1_1_1_1_3_5"/><span class="lab"><font color="Green" title="E78">A421</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_5"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c22">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E70">A398</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E69">A397</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_5"><span class="lab"><font color="Green" title="E75">A418</font></span></a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_5"/><span class="kw">then </span>
<font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_5"><span class="lab"><font color="Green" title="E77">A420</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_5"><span class="lab"><font color="Green" title="E78">A421</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E7:2_1_1_2_7_1_1_1_1_3_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_5"><span class="lab"><font color="Green" title="E76">A419</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E20:2_1_1_2_7_1_1_1_1_3"/><span class="kw">then </span>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font>
 ;<br/><a NAME="E21:2_1_1_2_7_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A422</font></span>: 
<font color="Maroon" title="c20">g2</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T9" onmouseover="rs('partfun2/T9')" onmouseout="rh()">PARTFUN2:9</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c22">lk</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/><a NAME="E22:2_1_1_2_7_1_1_1_1_3"/><span class="lab"><font color="Green" title="E76">A423</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><div><span class="lab"><font color="Green" title="E77">A424</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_6"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q</font> holds <br/><font color="Olive" title="b3">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> holds <br/><font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c25">p</font>, <font color="Maroon" title="c26">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">p</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q</font> implies <font color="Maroon" title="c25">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_3_6"/><span class="lab"><font color="Green" title="E78">A425</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_3_6"/><span class="lab"><font color="Green" title="E79">A426</font></span>: 
<font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_6"/><span class="lab"><font color="Green" title="E80">A427</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_6"/><span class="lab"><font color="Green" title="E81">A428</font></span>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">p</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_6"/><span class="lab"><font color="Green" title="E82">A429</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E70">A398</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E69">A397</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_6"><span class="lab"><font color="Green" title="E78">A425</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c27">n1</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font>;<br/><span class="kw">set </span><font color="Maroon" title="c28">m1</font> = <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font>;<br/><span class="kw">set </span><font color="Maroon" title="c29">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>);<br/><span class="kw">set </span><font color="Maroon" title="c30">pm</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span>);<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_6"/><span class="lab"><font color="Green" title="E83">A430</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_6"><span class="lab"><font color="Green" title="E80">A427</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_6"/>
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:2_1_1_2"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E70">A398</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E69">A397</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_6"><span class="lab"><font color="Green" title="E78">A425</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_6"/><span class="kw">then </span>
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_6"><span class="lab"><font color="Green" title="E82">A429</font></span></a>, <a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_3_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A431</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_6"><span class="lab"><font color="Green" title="E82">A429</font></span></a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_3_6"/><span class="lab"><font color="Green" title="E85">A432</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E70">A398</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E69">A397</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_6"><span class="lab"><font color="Green" title="E79">A426</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_3_6"/>
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:2_1_1_2"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E70">A398</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E69">A397</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_6"><span class="lab"><font color="Green" title="E79">A426</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_1_1_1_3_6"/><span class="kw">then </span>
<font color="Maroon" title="c10">l1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_6"><span class="lab"><font color="Green" title="E85">A432</font></span></a>, <a class="ref" href="matrix_0.html#D7" onmouseover="rs('matrix_0/D7')" onmouseout="rh()">MATRIX_0:def 7</a></span>;<br/><a NAME="E13:2_1_1_2_7_1_1_1_1_3_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A433</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_6"><span class="lab"><font color="Green" title="E85">A432</font></span></a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><a NAME="E14:2_1_1_2_7_1_1_1_1_3_6"/>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>) &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_6"><span class="lab"><font color="Green" title="E78">A425</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_6"><span class="lab"><font color="Green" title="E79">A426</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:2_1_1_2_7_1_1_1_1_3_6"/>
<font color="Maroon" title="c25">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:2_1_1_2"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_6"><span class="lab"><font color="Green" title="E81">A428</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_6"><span class="lab"><font color="Green" title="E82">A429</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_6"><span class="lab"><font color="Green" title="E85">A432</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_6"><span class="lab"><font color="Green" title="E83">A430</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_6"><span class="lab"><font color="Green" title="E84">A431</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3_6"><span class="lab"><font color="Green" title="E86">A433</font></span></a>, <a class="ref" href="seqm_3.html#D1" onmouseover="rs('seqm_3/D1')" onmouseout="rh()">SEQM_3:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E24:2_1_1_2_7_1_1_1_1_3"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> holds <br/> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b2">m</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> implies  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_3_7"/><span class="lab"><font color="Green" title="E78">A434</font></span>: 
<font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_3_7"/><span class="lab"><font color="Green" title="E79">A435</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_7"/><span class="lab"><font color="Green" title="E80">A436</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_7"/><span class="lab"><font color="Green" title="E81">A437</font></span>: 
<font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_7"/><span class="lab"><font color="Green" title="E82">A438</font></span>: 
<font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_7"/><span class="lab"><font color="Green" title="E83">A439</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D7" onmouseover="rs('xboole_0/D7')" onmouseout="rh()">XBOOLE_0:def 7</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c25">p1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c26">p2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <font color="Maroon" title="c27">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font>, <font color="Maroon" title="c28">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E8:2_1_1_2_7_1_1_1_1_3_7"/><span class="lab"><font color="Green" title="E84">A440</font></span>: 
( <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E73">A406</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E79">A435</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E80">A436</font></span></a></span>;<br/>
<a NAME="E9:2_1_1_2_7_1_1_1_1_3_7"/>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E10:2_1_1_2_7_1_1_1_1_3_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A441</font></span>: 
<font color="Maroon" title="c25">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E77">A424</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E79">A435</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E80">A436</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c29">lp</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c30">lq</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/>
<a NAME="E11:2_1_1_2_7_1_1_1_1_3_7"/><span class="lab"><font color="Green" title="E86">A442</font></span>: 
( <font color="Maroon" title="c25">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c26">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_1_1_1_3_7"/>
<font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_1_1_1_3_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A443</font></span>: 
<font color="Maroon" title="c27">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E77">A424</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E81">A437</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E82">A438</font></span></a></span>;<br/>
<a NAME="E14:2_1_1_2_7_1_1_1_1_3_7"/><span class="lab"><font color="Green" title="E88">A444</font></span>: 
( <font color="Maroon" title="c27">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c28">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c31">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span>;<br/>
<a NAME="E15:2_1_1_2_7_1_1_1_1_3_7"/><span class="lab"><font color="Green" title="E89">A445</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E83">A439</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E16:2_1_1_2_7_1_1_1_1_3_7"/><span class="lab"><font color="Green" title="E90">A446</font></span>: 
( <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E73">A406</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E81">A437</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E82">A438</font></span></a></span>;<br/>
<a NAME="E17:2_1_1_2_7_1_1_1_1_3_7"/><span class="lab"><font color="Green" title="E91">A447</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E83">A439</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E18:2_1_1_2_7_1_1_1_1_3_7"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E81">A437</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E82">A438</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c27">q1</font>,<font color="Maroon" title="c28">q2</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E87">A443</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E90">A446</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E88">A444</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>
;<br/>
<a NAME="E20:2_1_1_2_7_1_1_1_1_3_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A448</font></span>: 
 ex <font color="Olive" title="b1">tm</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">tm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> &amp; <font color="Olive" title="b1">tm</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">tm</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">tm</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E91">A447</font></span></a></span>;<br/>
<a NAME="E21:2_1_1_2_7_1_1_1_1_3_7"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E79">A435</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E80">A436</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c25">p1</font>,<font color="Maroon" title="c26">p2</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E85">A441</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E84">A440</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E86">A442</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>
;<br/>
<a NAME="E23:2_1_1_2_7_1_1_1_1_3_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A449</font></span>: 
 ex <font color="Olive" title="b1">tn</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">tn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> &amp; <font color="Olive" title="b1">tn</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">tn</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">tn</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E89">A445</font></span></a></span>;<br/>
<a NAME="E24:2_1_1_2_7_1_1_1_1_3_7"/>
<font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E77">A424</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E78">A434</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E80">A436</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E81">A437</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:2_1_1_2_7_1_1_1_1_3_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E93">A449</font></span></a>, <a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_3_7"><span class="lab"><font color="Green" title="E92">A448</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E25:2_1_1_2_7_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A450</font></span>: 
<font color="Maroon" title="c20">g2</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#T1" onmouseover="rs('goboard2/T1')" onmouseout="rh()">GOBOARD2:1</a></span>;<br/><a NAME="E26:2_1_1_2_7_1_1_1_1_3"/><span class="lab"><font color="Green" title="E79">A451</font></span>: 
 not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c23">ls</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span> } </span> ;<br/>
<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_3_8"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_8"/><span class="lab"><font color="Green" title="E80">A452</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c24">X</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_8"/><span class="lab"><font color="Green" title="E81">A453</font></span>: 
<font color="Maroon" title="c24">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_8"/><span class="lab"><font color="Green" title="E82">A454</font></span>: 
<font color="Maroon" title="c24">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c25">m</font>)
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_8"/><span class="lab"><font color="Green" title="E83">A455</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">m</font> &amp; <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_8"><span class="lab"><font color="Green" title="E81">A453</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">m</font>, <font color="Maroon" title="c27">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E9:2_1_1_2_7_1_1_1_1_3_8"/><span class="lab"><font color="Green" title="E84">A456</font></span>: 
<font color="Maroon" title="c25">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_8"><span class="lab"><font color="Green" title="E83">A455</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E10:2_1_1_2_7_1_1_1_1_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A457</font></span>: 
<font color="Maroon" title="c26">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E73">A406</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c28">lq</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/>
<a NAME="E11:2_1_1_2_7_1_1_1_1_3_8"/><span class="lab"><font color="Green" title="E86">A458</font></span>: 
( <font color="Maroon" title="c26">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c27">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_1_1_1_3_8"/><span class="lab"><font color="Green" title="E87">A459</font></span>: 
<font color="Maroon" title="c25">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_8"><span class="lab"><font color="Green" title="E83">A455</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_1_1_1_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A460</font></span>: 
<font color="Maroon" title="c27">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E73">A406</font></span></a></span>;<br/>
<a NAME="E14:2_1_1_2_7_1_1_1_1_3_8"/>
<font color="Maroon" title="c25">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E15:2_1_1_2_7_1_1_1_1_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A461</font></span>: 
<font color="Maroon" title="c26">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E77">A424</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_8"><span class="lab"><font color="Green" title="E84">A456</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3_8"><span class="lab"><font color="Green" title="E87">A459</font></span></a></span>;<br/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c25">m</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c26">q1</font>,<font color="Maroon" title="c27">q2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_8"><span class="lab"><font color="Green" title="E83">A455</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_8"><span class="lab"><font color="Green" title="E85">A457</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3_8"><span class="lab"><font color="Green" title="E88">A460</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_3_8"><span class="lab"><font color="Green" title="E89">A461</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3_8"><span class="lab"><font color="Green" title="E86">A458</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>
;<br/>
<a NAME="E17:2_1_1_2_7_1_1_1_1_3_8"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_8"><span class="lab"><font color="Green" title="E80">A452</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_8"><span class="lab"><font color="Green" title="E82">A454</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:2_1_1_2_7_1_1_1_1_3_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E73">A406</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_8"><span class="lab"><font color="Green" title="E84">A456</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E27:2_1_1_2_7_1_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c10">l1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E44:2_1_1_2"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E65">A393</font></span></a>, <a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><a NAME="E28:2_1_1_2_7_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A462</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E49:2_1_1_2"><span class="lab"><font color="Green" title="E52">A67</font></span></a>, <a class="txt" href="#E44:2_1_1_2"><span class="lab"><font color="Green" title="E47">A62</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E72">A405</font></span></a>, <a class="ref" href="seqm_3.html#D10" onmouseover="rs('seqm_3/D10')" onmouseout="rh()">SEQM_3:def 10</a></span>;<br/><div><span class="lab"><font color="Green" title="E81">A463</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_9"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> holds <br/> for <font color="Olive" title="b2">l1</font>, <font color="Olive" title="b3">l2</font>, <font color="Olive" title="b4">l3</font>, <font color="Olive" title="b5">l4</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">l1</font>,<font color="Olive" title="b3">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">l3</font>,<font color="Olive" title="b5">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">l1</font>,<font color="Olive" title="b3">l2</font>) &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b4">l3</font>,<font color="Olive" title="b5">l4</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> implies  for <font color="Olive" title="b1">l1</font>, <font color="Olive" title="b2">l2</font>, <font color="Olive" title="b3">l3</font>, <font color="Olive" title="b4">l4</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font>) &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_3_9"/><span class="lab"><font color="Green" title="E82">A464</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_3_9"/><span class="lab"><font color="Green" title="E83">A465</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">l1</font>, <font color="Olive" title="b2">l2</font>, <font color="Olive" title="b3">l3</font>, <font color="Olive" title="b4">l4</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font>) &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c24">l1</font>, <font color="Maroon" title="c25">l2</font>, <font color="Maroon" title="c26">l3</font>, <font color="Maroon" title="c27">l4</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">l1</font>,<font color="Maroon" title="c25">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">l3</font>,<font color="Maroon" title="c27">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c24">l1</font>,<font color="Maroon" title="c25">l2</font>) &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c26">l3</font>,<font color="Maroon" title="c27">l4</font>) implies <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_9"/><span class="lab"><font color="Green" title="E84">A466</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">l1</font>,<font color="Maroon" title="c25">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_9"/><span class="lab"><font color="Green" title="E85">A467</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">l3</font>,<font color="Maroon" title="c27">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_9"/><span class="lab"><font color="Green" title="E86">A468</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c24">l1</font>,<font color="Maroon" title="c25">l2</font>)
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_9"/><span class="lab"><font color="Green" title="E87">A469</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c26">l3</font>,<font color="Maroon" title="c27">l4</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_9"/>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E70">A398</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E69">A397</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_9"><span class="lab"><font color="Green" title="E83">A465</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A470</font></span>: 
( <font color="Maroon" title="c26">l3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> &amp; <font color="Maroon" title="c27">l4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_9"><span class="lab"><font color="Green" title="E85">A467</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_9"><span class="lab"><font color="Green" title="E87">A469</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_3_9"/>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E70">A398</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E69">A397</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_9"><span class="lab"><font color="Green" title="E82">A464</font></span></a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_3_9"/><span class="kw">then </span>
( <font color="Maroon" title="c24">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> &amp; <font color="Maroon" title="c25">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_9"><span class="lab"><font color="Green" title="E84">A466</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_9"><span class="lab"><font color="Green" title="E86">A468</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> = 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_9"><span class="lab"><font color="Green" title="E88">A470</font></span></a>, <a class="ref" href="absvalue.html#T2" onmouseover="rs('absvalue/T2')" onmouseout="rh()">ABSVALUE:2</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>

<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default">1</span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T52" onmouseover="rs('complex1/T52')" onmouseout="rh()">COMPLEX1:52</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_10"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">l1</font>, <font color="Olive" title="b2">l2</font>, <font color="Olive" title="b3">l3</font>, <font color="Olive" title="b4">l4</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font>) &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font>) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font> &amp; 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">l1</font>, <font color="Maroon" title="c24">l2</font>, <font color="Maroon" title="c25">l3</font>, <font color="Maroon" title="c26">l4</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">l1</font>,<font color="Maroon" title="c24">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">l3</font>,<font color="Maroon" title="c26">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c23">l1</font>,<font color="Maroon" title="c24">l2</font>) &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">l3</font>,<font color="Maroon" title="c26">l4</font>) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font> &amp; 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> implies <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c25">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_3_10"/><span class="lab"><font color="Green" title="E82">A471</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">l1</font>,<font color="Maroon" title="c24">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_3_10"/><span class="lab"><font color="Green" title="E83">A472</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">l3</font>,<font color="Maroon" title="c26">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_10"/><span class="lab"><font color="Green" title="E84">A473</font></span>: 
<font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c23">l1</font>,<font color="Maroon" title="c24">l2</font>)
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_10"/><span class="lab"><font color="Green" title="E85">A474</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">l3</font>,<font color="Maroon" title="c26">l4</font>)
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_10"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_10"/><span class="lab"><font color="Green" title="E86">A475</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c25">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_10"/>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E70">A398</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E69">A397</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_10"><span class="lab"><font color="Green" title="E86">A475</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A476</font></span>: 
( <font color="Maroon" title="c25">l3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> &amp; <font color="Maroon" title="c26">l4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_10"><span class="lab"><font color="Green" title="E83">A472</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_10"><span class="lab"><font color="Green" title="E85">A474</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_3_10"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_3_10"/><span class="kw">then </span>
( <font color="Maroon" title="c23">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> &amp; <font color="Maroon" title="c24">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E20:2_1_1_2"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_10"><span class="lab"><font color="Green" title="E82">A471</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_10"><span class="lab"><font color="Green" title="E84">A473</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c25">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> = 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_10"><span class="lab"><font color="Green" title="E87">A476</font></span></a>, <a class="ref" href="absvalue.html#T2" onmouseover="rs('absvalue/T2')" onmouseout="rh()">ABSVALUE:2</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">i2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>

<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default">1</span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T52" onmouseover="rs('complex1/T52')" onmouseout="rh()">COMPLEX1:52</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E31:2_1_1_2_7_1_1_1_1_3"/><span class="kw">then </span>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/> for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font>) &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">i</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">j</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:2_1_1_2"><span class="lab"><font color="Green" title="E40">A48</font></span></a>, <a class="txt" href="#E29:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E81">A463</font></span></a>, <a class="ref" href="goboard1.html#T24" onmouseover="rs('goboard1/T24')" onmouseout="rh()">GOBOARD1:24</a></span>;<br/><span class="kw">hence </span><a NAME="E32:2_1_1_2_7_1_1_1_1_3"/>
<font color="Maroon" title="c21">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E71">A404</font></span></a>, <a class="ref" href="goboard1.html#D9" onmouseover="rs('goboard1/D9')" onmouseout="rh()">GOBOARD1:def 9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )</span><br/><a NAME="E33:2_1_1_2_7_1_1_1_1_3"/><span class="lab"><font color="Green" title="E82">A477</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><span class="lab"><font color="Green" title="E83">A478</font></span>:  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E12:2_1_1_2"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E67">A395</font></span></a>, <a class="txt" href="#E28:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E80">A462</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E76">A423</font></span></a>, <a class="txt" href="#E33:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E82">A477</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>
;<br/><a NAME="E35:2_1_1_2_7_1_1_1_1_3"/><span class="lab"><font color="Green" title="E84">A479</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font> or <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_3_12"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_12"/><span class="lab"><font color="Green" title="E85">A480</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_12"/><span class="lab"><font color="Green" title="E86">A481</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">n</font></span>)</span>);<br/>
<a NAME="E5:2_1_1_2_7_1_1_1_1_3_12"/><span class="lab"><font color="Green" title="E87">A482</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">n</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_12"><span class="lab"><font color="Green" title="E85">A480</font></span></a></span>;<br/>
<a NAME="E6:2_1_1_2_7_1_1_1_1_3_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A483</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">n</font></span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E73">A406</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_12"><span class="lab"><font color="Green" title="E85">A480</font></span></a></span>;<br/>
<a NAME="E7:2_1_1_2_7_1_1_1_1_3_12"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">n</font></span>)</span>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">n</font></span>)</span>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E73">A406</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_12"><span class="lab"><font color="Green" title="E85">A480</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_12"><span class="lab"><font color="Green" title="E87">A482</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:2_1_1_2_7_1_1_1_1_3_12"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E83">A478</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_12"><span class="lab"><font color="Green" title="E86">A481</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_12"><span class="lab"><font color="Green" title="E87">A482</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_12"><span class="lab"><font color="Green" title="E88">A483</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E36:2_1_1_2_7_1_1_1_1_3"/><span class="lab"><font color="Green" title="E85">A484</font></span>: 
 not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_3_13"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_13"/><span class="lab"><font color="Green" title="E86">A485</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_13"/><span class="lab"><font color="Green" title="E87">A486</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E5:2_1_1_2_7_1_1_1_1_3_13"/><span class="lab"><font color="Green" title="E88">A487</font></span>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_13"><span class="lab"><font color="Green" title="E86">A485</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:2_1_1_2_7_1_1_1_1_3_13"/><span class="lab"><font color="Green" title="E89">A488</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">n</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_13"><span class="lab"><font color="Green" title="E86">A485</font></span></a></span>;<br/>
<a NAME="E7:2_1_1_2_7_1_1_1_1_3_13"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E8:2_1_1_2_7_1_1_1_1_3_13"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E70">A398</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_13"><span class="lab"><font color="Green" title="E86">A485</font></span></a></span>;<br/>
<a NAME="E9:2_1_1_2_7_1_1_1_1_3_13"/><span class="kw">then </span>
<font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_13"><span class="lab"><font color="Green" title="E87">A486</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_13"><span class="lab"><font color="Green" title="E89">A488</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:2_1_1_2_7_1_1_1_1_3_13"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_13"><span class="lab"><font color="Green" title="E88">A487</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E37:2_1_1_2_7_1_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_14"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c23">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_3_14"/><span class="lab"><font color="Green" title="E86">A489</font></span>: 
 not <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:2_1_1_2_7_1_1_1_1_3_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A490</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:2_1_1_2_7_1_1_1_1_3_14"/><span class="lab"><font color="Green" title="E88">A491</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_14"><span class="lab"><font color="Green" title="E86">A489</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_14_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_14_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_3_14_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_14_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_14_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_1_2_7_1_1_1_1_3_14_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E36:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E85">A484</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_14"><span class="lab"><font color="Green" title="E88">A491</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_14"><span class="lab"><font color="Green" title="E87">A490</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_14_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_14_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_14_1_1_2"/><span class="kw">then </span>
(  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E38:2_1_1_2"><span class="lab"><font color="Green" title="E41">A49</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A479</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_14"><span class="lab"><font color="Green" title="E88">A491</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_14"><span class="lab"><font color="Green" title="E87">A490</font></span></a>, <a class="ref" href="goboard2.html#T4" onmouseover="rs('goboard2/T4')" onmouseout="rh()">GOBOARD2:4</a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_1_1_1_3_14_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E85">A484</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_14"><span class="lab"><font color="Green" title="E87">A490</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_1_1_1_3_14"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E38:2_1_1_2_7_1_1_1_1_3"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 ;<br/><span class="kw">hence </span><a NAME="E39:2_1_1_2_7_1_1_1_1_3"/>
<font color="Maroon" title="c21">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:2_1_1_2"><span class="lab"><font color="Green" title="E32">A40</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E75">A422</font></span></a>, <a class="ref" href="finseq_3.html#T91" onmouseover="rs('finseq_3/T91')" onmouseout="rh()">FINSEQ_3:91</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )</span><br/><a NAME="E40:2_1_1_2_7_1_1_1_1_3"/><span class="lab"><font color="Green" title="E86">A492</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E12:2_1_1_2"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1"><span class="lab"><font color="Green" title="E61">A76</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E41:2_1_1_2_7_1_1_1_1_3"/><span class="lab"><font color="Green" title="E87">A493</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> holds <br/><span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_15"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> implies <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_3_15"/><span class="lab"><font color="Green" title="E88">A494</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_3_15"/><span class="lab"><font color="Green" title="E89">A495</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E3:2_1_1_2_7_1_1_1_1_3_15"/><span class="lab"><font color="Green" title="E90">A496</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E88">A494</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E89">A495</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/>
<a NAME="E4:2_1_1_2_7_1_1_1_1_3_15"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E5:2_1_1_2_7_1_1_1_1_3_15"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E83">A478</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A479</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_15"/><span class="lab"><font color="Green" title="E91">A497</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">u1</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_15"/><span class="lab"><font color="Green" title="E92">A498</font></span>: 
<font color="Maroon" title="c24">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E9:2_1_1_2_7_1_1_1_1_3_15"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E10:2_1_1_2_7_1_1_1_1_3_15"/>
<font color="Maroon" title="c24">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/>
<a NAME="E11:2_1_1_2_7_1_1_1_1_3_15"/><span class="lab"><font color="Green" title="E93">A499</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E88">A494</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E89">A495</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_1_1_1_3_15"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_1_1_1_3_15"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E83">A478</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A479</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">u2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E15:2_1_1_2_7_1_1_1_1_3_15"/><span class="lab"><font color="Green" title="E94">A500</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u2</font>
 <span class="kw">and </span><br/><a NAME="E16:2_1_1_2_7_1_1_1_1_3_15"/><span class="lab"><font color="Green" title="E95">A501</font></span>: 
<font color="Maroon" title="c25">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E17:2_1_1_2_7_1_1_1_1_3_15"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E18:2_1_1_2_7_1_1_1_1_3_15"/>
<font color="Maroon" title="c25">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/>
<a NAME="E19:2_1_1_2_7_1_1_1_1_3_15"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E20:2_1_1_2_7_1_1_1_1_3_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A502</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c24">u1</font>,<font color="Maroon" title="c25">u2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E89">A495</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E91">A497</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E94">A500</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E21:2_1_1_2_7_1_1_1_1_3_15"/>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E22:2_1_1_2_7_1_1_1_1_3_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A503</font></span>: 
<font color="Maroon" title="c24">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E77">A424</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E90">A496</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E93">A499</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E91">A497</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E94">A500</font></span></a></span>;<br/>
<a NAME="E23:2_1_1_2_7_1_1_1_1_3_15"/><span class="lab"><font color="Green" title="E98">A504</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E88">A494</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E89">A495</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/>
<a NAME="E24:2_1_1_2_7_1_1_1_1_3_15"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E25:2_1_1_2_7_1_1_1_1_3_15"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E83">A478</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A479</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E27:2_1_1_2_7_1_1_1_1_3_15"/><span class="lab"><font color="Green" title="E99">A505</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u</font>
 <span class="kw">and </span><br/><a NAME="E28:2_1_1_2_7_1_1_1_1_3_15"/><span class="lab"><font color="Green" title="E100">A506</font></span>: 
<font color="Maroon" title="c26">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E29:2_1_1_2_7_1_1_1_1_3_15"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E30:2_1_1_2_7_1_1_1_1_3_15"/>
<font color="Maroon" title="c26">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/>
<a NAME="E31:2_1_1_2_7_1_1_1_1_3_15"/>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E32:2_1_1_2_7_1_1_1_1_3_15"/><span class="kw">then </span>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E89">A495</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E33:2_1_1_2_7_1_1_1_1_3_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A507</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c26">u</font>,<font color="Maroon" title="c24">u1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E88">A494</font></span></a>, <a class="txt" href="#E27:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E99">A505</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E91">A497</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c27">lg</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c26">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/>
<a NAME="E34:2_1_1_2_7_1_1_1_1_3_15"/>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E35:2_1_1_2_7_1_1_1_1_3_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A508</font></span>: 
<font color="Maroon" title="c26">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E77">A424</font></span></a>, <a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E98">A504</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E90">A496</font></span></a>, <a class="txt" href="#E27:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E99">A505</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E91">A497</font></span></a></span>;<br/>
<a NAME="E36:2_1_1_2_7_1_1_1_1_3_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A509</font></span>: 
<font color="Maroon" title="c24">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c26">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E92">A498</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E97">A503</font></span></a></span>;<br/>
<a NAME="E37:2_1_1_2_7_1_1_1_1_3_15"/>
( <font color="Maroon" title="c26">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c25">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<a NAME="E38:2_1_1_2_7_1_1_1_1_3_15"/><span class="kw">then </span>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c26">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E99">A505</font></span></a>, <a class="txt" href="#E28:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E100">A506</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E94">A500</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E95">A501</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E97">A503</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E102">A508</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E39:2_1_1_2_7_1_1_1_1_3_15"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E99">A505</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E91">A497</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E94">A500</font></span></a>, <a class="txt" href="#E33:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E101">A507</font></span></a>, <a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E96">A502</font></span></a>, <a class="txt" href="#E36:2_1_1_2_7_1_1_1_1_3_15"><span class="lab"><font color="Green" title="E103">A509</font></span></a>, <a class="ref" href="topreal1.html#T8" onmouseover="rs('topreal1/T8')" onmouseout="rh()">TOPREAL1:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E42:2_1_1_2_7_1_1_1_1_3"/>
<font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_16"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topreal1.html#D6" onmouseover="rs('topreal1/D6')" onmouseout="rh()">TOPREAL1:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> or  not <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> or <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_3_16"/><span class="lab"><font color="Green" title="E88">A510</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_3_16"/><span class="lab"><font color="Green" title="E89">A511</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E3:2_1_1_2_7_1_1_1_1_3_16"/><span class="lab"><font color="Green" title="E90">A512</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E89">A511</font></span></a></span>;<br/>
<a NAME="E4:2_1_1_2_7_1_1_1_1_3_16"/>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E5:2_1_1_2_7_1_1_1_1_3_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A513</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E89">A511</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E6:2_1_1_2_7_1_1_1_1_3_16"/><span class="lab"><font color="Green" title="E92">A514</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<a NAME="E7:2_1_1_2_7_1_1_1_1_3_16"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2
 
;<br/>
<a NAME="E8:2_1_1_2_7_1_1_1_1_3_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A515</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E89">A511</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E92">A514</font></span></a>, <a class="ref" href="xreal_1.html#T20" onmouseover="rs('xreal_1/T20')" onmouseout="rh()">XREAL_1:20</a></span>;<br/>
<a NAME="E9:2_1_1_2_7_1_1_1_1_3_16"/><span class="lab"><font color="Green" title="E94">A516</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E10:2_1_1_2_7_1_1_1_1_3_16"/><span class="lab"><font color="Green" title="E95">A517</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E11:2_1_1_2_7_1_1_1_1_3_16"/><span class="lab"><font color="Green" title="E96">A518</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_16_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_3_16_1"/>
( <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_16_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_16_1_1"/><span class="lab"><font color="Green" title="E97">A519</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_16_1_1"/><span class="lab"><font color="Green" title="E98">A520</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_16_1_1"/><span class="lab"><font color="Green" title="E99">A521</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E88">A510</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E97">A519</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_16_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A522</font></span>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_16_1_1"/>
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E88">A510</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E97">A519</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_16_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A523</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c23">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E99">A521</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_16_1_1"/>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E88">A510</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E97">A519</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_16_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E99">A521</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E98">A520</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_1_1_1_3_16_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_1_2"><span class="lab"><font color="Green" title="E33">A41</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E88">A510</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E97">A519</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E101">A523</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E100">A522</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_16_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_16_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_16_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_16_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A524</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_16_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_16_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/><span class="lab"><font color="Green" title="E98">A525</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><a NAME="E1:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E88">A510</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E98">A525</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E2:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A526</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">assume </span><a NAME="E3:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E4:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E99">A526</font></span></a>, <a class="ref" href="topreal1.html#T23" onmouseover="rs('topreal1/T23')" onmouseout="rh()">TOPREAL1:23</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A527</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="goboard2.html#T4" onmouseover="rs('goboard2/T4')" onmouseout="rh()">GOBOARD2:4</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/>
( <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>) &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E88">A510</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E89">A511</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E94">A516</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E91">A513</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E96">A518</font></span></a>, <a class="ref" href="topreal1.html#T21" onmouseover="rs('topreal1/T21')" onmouseout="rh()">TOPREAL1:21</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A528</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/><span class="lab"><font color="Green" title="E101">A529</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E90">A512</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E98">A525</font></span></a>, <a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E92">A514</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A530</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E83">A478</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A479</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E13:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/><span class="lab"><font color="Green" title="E103">A531</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">u1</font>
 <span class="kw">and </span><br/><a NAME="E14:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/>
<font color="Maroon" title="c24">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E15:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E16:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/>
<font color="Maroon" title="c24">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/><a NAME="E17:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/>
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T68" onmouseover="rs('rltopsp1/T68')" onmouseout="rh()">RLTOPSP1:68</a></span>;<br/><a NAME="E18:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A532</font></span>: 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<font color="Maroon" title="c24">u1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E86">A492</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A479</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A530</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E103">A531</font></span></a>, <a class="ref" href="topreal1.html#T6" onmouseover="rs('topreal1/T6')" onmouseout="rh()">TOPREAL1:6</a></span>;<br/><a NAME="E19:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E20:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A533</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E98">A525</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A534</font></span>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E98">A525</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/><a NAME="E22:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/>
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E88">A510</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E95">A517</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E98">A525</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E23:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A535</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c23">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A533</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E24:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/>
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E96">A518</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E92">A514</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E98">A525</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A529</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><a NAME="E25:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A536</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<font color="Maroon" title="c24">u1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E89">A511</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E94">A516</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E96">A518</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E106">A534</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E103">A531</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E26:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c23">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="ref" href="topreal3.html#T19" onmouseover="rs('topreal3/T19')" onmouseout="rh()">TOPREAL3:19</a></span>;<br/><a NAME="E27:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E99">A527</font></span></a>, <a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E107">A535</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E106">A534</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E104">A532</font></span></a>, <a class="txt" href="#E25:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E108">A536</font></span></a>, <a class="ref" href="xboole_1.html#T27" onmouseover="rs('xboole_1/T27')" onmouseout="rh()">XBOOLE_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E28:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A528</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_16_1_2"><span class="lab"><font color="Green" title="E97">A524</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A537</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">n1</font> = <font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E99">A538</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A539</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E89">A511</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E92">A514</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A540</font></span>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E99">A538</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A539</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A541</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E83">A478</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A479</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E10:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E103">A542</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u1</font>
 <span class="kw">and </span><br/><a NAME="E11:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E104">A543</font></span>: 
( <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">and </span><br/><a NAME="E12:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c25">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/><a NAME="E13:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E14:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E99">A538</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/><a NAME="E16:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A544</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<font color="Maroon" title="c25">u1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E88">A510</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E91">A513</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A540</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E103">A542</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E17:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E106">A545</font></span>: 
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A539</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E18:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/><a NAME="E19:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E83">A478</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A479</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">u2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E21:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E107">A546</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u2</font>
 <span class="kw">and </span><br/><a NAME="E22:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E108">A547</font></span>: 
( <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">and </span><br/><a NAME="E23:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c26">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/><span class="kw">set </span><font color="Maroon" title="c27">lg</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/><a NAME="E24:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c25">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E77">A424</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A541</font></span></a>, <a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E106">A545</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E103">A542</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E107">A546</font></span></a></span>;<br/><a NAME="E25:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c26">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c25">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E104">A543</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><a NAME="E26:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A548</font></span>: 
<font color="Maroon" title="c25">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<font color="Maroon" title="c26">u2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E76">A423</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E108">A547</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>;<br/><a NAME="E27:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E99">A538</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A539</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><a NAME="E28:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c25">u1</font>,<font color="Maroon" title="c26">u2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E89">A511</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E94">A516</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E96">A518</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A540</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E103">A542</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E107">A546</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E29:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A540</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E103">A542</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A544</font></span></a>, <a class="txt" href="#E26:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E109">A548</font></span></a>, <a class="ref" href="topreal1.html#T8" onmouseover="rs('topreal1/T8')" onmouseout="rh()">TOPREAL1:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A549</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2"><span class="lab"><font color="Green" title="E98">A537</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A550</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font>
 ;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A551</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">n1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E91">A513</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E99">A549</font></span></a>, <a class="ref" href="goboard2.html#T5" onmouseover="rs('goboard2/T5')" onmouseout="rh()">GOBOARD2:5</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E102">A552</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E91">A513</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E92">A514</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A553</font></span>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E102">A552</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E95">A517</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E99">A549</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E90">A512</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E102">A552</font></span></a>, <a class="ref" href="goboard2.html#T5" onmouseover="rs('goboard2/T5')" onmouseout="rh()">GOBOARD2:5</a></span>;<br/><span class="kw">hence </span><a NAME="E12:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E87">A493</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_16"><span class="lab"><font color="Green" title="E93">A515</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E101">A551</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E103">A553</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E100">A550</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_1_1_1_3_16_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_1_1_1_3_16_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E43:2_1_1_2_7_1_1_1_1_3"/><span class="lab"><font color="Green" title="E88">A554</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_17"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font> or <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c24">ls</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span> } </span> ;<br/>
<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_3_17"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_17"/><span class="lab"><font color="Green" title="E89">A555</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">X</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_17"/><span class="lab"><font color="Green" title="E90">A556</font></span>: 
<font color="Maroon" title="c25">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c26">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_17"/><span class="lab"><font color="Green" title="E91">A557</font></span>: 
<font color="Maroon" title="c25">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c26">m</font>)
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_17"/><span class="lab"><font color="Green" title="E92">A558</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">m</font> &amp; <font color="Maroon" title="c26">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_17"><span class="lab"><font color="Green" title="E90">A556</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">m</font>, <font color="Maroon" title="c28">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E9:2_1_1_2_7_1_1_1_1_3_17"/><span class="lab"><font color="Green" title="E93">A559</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c26">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c27">q1</font>,<font color="Maroon" title="c28">q2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_17"><span class="lab"><font color="Green" title="E92">A558</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E10:2_1_1_2_7_1_1_1_1_3_17"/>
<font color="Maroon" title="c26">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_17"><span class="lab"><font color="Green" title="E92">A558</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E11:2_1_1_2_7_1_1_1_1_3_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A560</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_1_1_1_3_17"/>
<font color="Maroon" title="c26">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_17"><span class="lab"><font color="Green" title="E92">A558</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_1_1_1_3_17"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E14:2_1_1_2_7_1_1_1_1_3_17"/><span class="kw">then </span>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c27">q1</font>,<font color="Maroon" title="c28">q2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E86">A492</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E84">A479</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3_17"><span class="lab"><font color="Green" title="E94">A560</font></span></a>, <a class="ref" href="topreal1.html#T6" onmouseover="rs('topreal1/T6')" onmouseout="rh()">TOPREAL1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:2_1_1_2_7_1_1_1_1_3_17"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E86">A492</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_17"><span class="lab"><font color="Green" title="E89">A555</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_17"><span class="lab"><font color="Green" title="E91">A557</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_17"><span class="lab"><font color="Green" title="E93">A559</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E44:2_1_1_2_7_1_1_1_1_3"/><span class="lab"><font color="Green" title="E89">A561</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_18"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_18_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_3_18_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_18_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_18_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_1_2_7_1_1_1_1_3_18_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2"><span class="lab"><font color="Green" title="E43">A52</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_18_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_18_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_18_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_18_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="ref" href="goboard2.html#T4" onmouseover="rs('goboard2/T4')" onmouseout="rh()">GOBOARD2:4</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_18_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A562</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E88">A554</font></span></a>, <a class="ref" href="xboole_1.html#T26" onmouseover="rs('xboole_1/T26')" onmouseout="rh()">XBOOLE_1:26</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_18_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c23">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span>;<br/><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_3_18_1_2_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_3_18_1_2_1"/><span class="kw">then </span>
(  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_18_1_2"><span class="lab"><font color="Green" title="E90">A562</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_1_1_1_3_18_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E79">A451</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_1_1_1_3_18_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E45:2_1_1_2_7_1_1_1_1_3"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Olive" title="b2">m</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E90">A563</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E2:2_1_1_2_7_1_1_1_1_3_19"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E91">A564</font></span>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> = 
<font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">qq</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c24">l1</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">l2</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span> } </span> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c26">n</font>, <font color="Maroon" title="c27">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c27">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> implies  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E92">A565</font></span>: 
<font color="Maroon" title="c27">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E93">A566</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E94">A567</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E95">A568</font></span>: 
<font color="Maroon" title="c27">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">and </span><br/><a NAME="E9:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E96">A569</font></span>: 
<font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/>

<a NAME="E10:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E97">A570</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E93">A566</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E11:2_1_1_2_7_1_1_1_1_3_19"/>
<font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E62">A390</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_1_1_1_3_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A571</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_1_1_1_3_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A572</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E14:2_1_1_2_7_1_1_1_1_3_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A573</font></span>: 
( <font color="Maroon" title="c23">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c23">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E73">A406</font></span></a></span>;<br/>
<a NAME="E15:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E101">A574</font></span>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">qq</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E98">A571</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/>
<a NAME="E16:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E102">A575</font></span>: 
<font color="Maroon" title="c23">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E73">A406</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E99">A572</font></span></a></span>;<br/>
<a NAME="E17:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E103">A576</font></span>: 
<font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E96">A569</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E18:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E104">A577</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E96">A569</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E19:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E105">A578</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E94">A567</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E20:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E106">A579</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E94">A567</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E21:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E107">A580</font></span>: 
<font color="Maroon" title="c23">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<a NAME="E22:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E108">A581</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E95">A568</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c28">ql</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/>
<a NAME="E23:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E109">A582</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E24:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E110">A583</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<a NAME="E25:2_1_1_2_7_1_1_1_1_3_19"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E98">A571</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E111">A584</font></span>:  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<font color="Maroon" title="c23">qq</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E90">A563</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E91">A564</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E101">A574</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E76">A423</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E100">A573</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E107">A580</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>
;<br/>
<a NAME="E27:2_1_1_2_7_1_1_1_1_3_19"/><span class="lab"><font color="Green" title="E112">A585</font></span>: 
<font color="Maroon" title="c27">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E28:2_1_1_2_7_1_1_1_1_3_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A586</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E92">A565</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_3_19_3_1"/>
( <font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_19_3_1_1"/><span class="lab"><font color="Green" title="E114">A587</font></span>: 
<font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_19_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E112">A585</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_19_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A588</font></span>: 
<font color="Maroon" title="c27">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E108">A581</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_19_3_1_1"/>
<font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E104">A577</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_19_3_1_1"><span class="lab"><font color="Green" title="E114">A587</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_19_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A589</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c27">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_19_3_1_1"><span class="lab"><font color="Green" title="E115">A588</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_19_3_1_1"/><span class="lab"><font color="Green" title="E117">A590</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E113">A586</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_19_3_1_1"><span class="lab"><font color="Green" title="E114">A587</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_19_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c26">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E109">A582</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_19_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A591</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E97">A570</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_3_19_3_1_1"/>
<font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E105">A578</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_19_3_1_1"><span class="lab"><font color="Green" title="E117">A590</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_3_19_3_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c26">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_19_3_1_1"><span class="lab"><font color="Green" title="E118">A591</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><span class="kw">hence </span><a NAME="E11:2_1_1_2_7_1_1_1_1_3_19_3_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:2_1_1_2"><span class="lab"><font color="Green" title="E34">A42</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E92">A565</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_19_3_1_1"><span class="lab"><font color="Green" title="E116">A589</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_19_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A592</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">m1</font> = <font color="Maroon" title="c27">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1"/>
( <font color="Maroon" title="c27">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or <font color="Maroon" title="c27">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E115">A593</font></span>: 
<font color="Maroon" title="c27">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E116">A594</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>) or <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">px</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E117">A595</font></span>: 
( <font color="Maroon" title="c31">px</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">x</font> &amp; <font color="Maroon" title="c31">px</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">px</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E118">A596</font></span>: 
<font color="Maroon" title="c31">px</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E111">A584</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E115">A593</font></span></a></span>;<br/>
<a NAME="E5:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c31">px</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E102">A575</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E118">A596</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E83">A478</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E117">A595</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"/>
<font color="Maroon" title="c26">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E92">A565</font></span></a>, <a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E109">A582</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E115">A593</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A597</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E97">A570</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><a NAME="E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E92">A565</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E105">A578</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E115">A593</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E2:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A598</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">assume </span><a NAME="E3:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E118">A598</font></span></a>, <a class="ref" href="topreal1.html#T23" onmouseover="rs('topreal1/T23')" onmouseout="rh()">TOPREAL1:23</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A599</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="goboard2.html#T4" onmouseover="rs('goboard2/T4')" onmouseout="rh()">GOBOARD2:4</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E119">A600</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E92">A565</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E105">A578</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E115">A593</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A601</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c26">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E117">A597</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E92">A565</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E97">A570</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E115">A593</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A602</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E118">A599</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E116">A594</font></span></a>, <a class="ref" href="xboole_1.html#T27" onmouseover="rs('xboole_1/T27')" onmouseout="rh()">XBOOLE_1:27</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c30">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span>;<br/><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_3"/><span class="lab"><font color="Green" title="E122">A603</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A604</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_3"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E121">A602</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E122">A603</font></span></a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A605</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_3"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>;<br/><span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:2_1_1_2"><span class="lab"><font color="Green" title="E32">A40</font></span></a>, <a class="txt" href="#E30:2_1_1_2"><span class="lab"><font color="Green" title="E33">A41</font></span></a>, <a class="txt" href="#E31:2_1_1_2"><span class="lab"><font color="Green" title="E34">A42</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E92">A565</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E115">A593</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E117">A597</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E119">A600</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E120">A601</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E123">A604</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E124">A605</font></span></a>, <a class="ref" href="goboard2.html#T2" onmouseover="rs('goboard2/T2')" onmouseout="rh()">GOBOARD2:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E14:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"/>
<font color="Maroon" title="c27">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A606</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_19_3_1_2"><span class="lab"><font color="Green" title="E114">A592</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A607</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"/>
<font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A608</font></span>: 
<font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E103">A576</font></span></a>, <a class="txt" href="#E24:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E110">A583</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"/>
<font color="Maroon" title="c27">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A609</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c29">m1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E103">A576</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E115">A606</font></span></a>, <a class="ref" href="goboard2.html#T5" onmouseover="rs('goboard2/T5')" onmouseout="rh()">GOBOARD2:5</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A607</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E117">A608</font></span></a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A610</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E120">A611</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c26">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E109">A582</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A612</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E97">A570</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E105">A578</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E120">A611</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c26">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E121">A612</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E97">A570</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E120">A611</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E89">A561</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E119">A610</font></span></a>, <a class="ref" href="xboole_1.html#T3" onmouseover="rs('xboole_1/T3')" onmouseout="rh()">XBOOLE_1:3</a>, <a class="ref" href="xboole_1.html#T27" onmouseover="rs('xboole_1/T27')" onmouseout="rh()">XBOOLE_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A613</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">n1</font> = <font color="Maroon" title="c26">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E121">A614</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E122">A615</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">n</font> or <font color="Maroon" title="c26">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E123">A616</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c31">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">m1</font>, <font color="Maroon" title="c32">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c33">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c34">q1l</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c31">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">v</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/><a NAME="E2:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E124">A617</font></span>: 
( <font color="Maroon" title="c31">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c32">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c32">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><span class="kw">assume </span><a NAME="E3:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E125">A618</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A619</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E125">A618</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A620</font></span>: 
 ex <font color="Olive" title="b1">qx</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">qx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> &amp; <font color="Olive" title="b1">qx</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qx</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">qx</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E111">A584</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E123">A616</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E128">A621</font></span>: 
<font color="Maroon" title="c29">m1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A607</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E117">A608</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A622</font></span>: 
<font color="Maroon" title="c31">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E73">A406</font></span></a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E130">A623</font></span>: 
<font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A607</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E117">A608</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A624</font></span>: 
<font color="Maroon" title="c32">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E73">A406</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A625</font></span>: 
<font color="Maroon" title="c31">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c32">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E77">A424</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E128">A621</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E130">A623</font></span></a></span>;<br/> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c29">m1</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c31">q1</font>,<font color="Maroon" title="c32">q2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A607</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E117">A608</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c31">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">v</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E129">A622</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E131">A624</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E132">A625</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E124">A617</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>
;<br/><a NAME="E14:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A626</font></span>: 
 ex <font color="Olive" title="b1">qy</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">qy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> &amp; <font color="Olive" title="b1">qy</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c31">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qy</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">qy</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A609</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E126">A619</font></span></a></span>;<br/><a NAME="E15:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E92">A565</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E121">A614</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E16:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E17:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E77">A424</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E99">A572</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E128">A621</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E18:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E127">A620</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E133">A626</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c26">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A613</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A627</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c30">n1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E106">A579</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E122">A615</font></span></a>, <a class="ref" href="goboard2.html#T5" onmouseover="rs('goboard2/T5')" onmouseout="rh()">GOBOARD2:5</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_19"><span class="lab"><font color="Green" title="E92">A565</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E121">A614</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E78">A450</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A609</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E123">A627</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E12:2_1_1_2_7_1_1_1_1_3_19_3_1_2_1_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_1_1_1_3_19_3_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E30:2_1_1_2_7_1_1_1_1_3_19"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E46:2_1_1_2_7_1_1_1_1_3"/>
<font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#T1" onmouseover="rs('goboard2/T1')" onmouseout="rh()">GOBOARD2:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_20"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c23">p</font> = <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c24">q</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1;<br/><a NAME="E1:2_1_1_2_7_1_1_1_1_3_20"/>
<font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E62">A390</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E2:2_1_1_2_7_1_1_1_1_3_20"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_20"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_20"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E73">A406</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_1_1_1_3_20"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E48:2_1_1_2_7_1_1_1_1_3"/>
<font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:2_1_1_2"><span class="lab"><font color="Green" title="E35">A43</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E74">A415</font></span></a>, <a class="ref" href="goboard2.html#T8" onmouseover="rs('goboard2/T8')" onmouseout="rh()">GOBOARD2:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )</span><br/><span class="kw">thus </span><a NAME="E49:2_1_1_2_7_1_1_1_1_3"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c23">lg</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c24">lf</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span> } </span> ;<br/>
<a NAME="E1:2_1_1_2_7_1_1_1_1_3_21"/><span class="lab"><font color="Green" title="E90">A628</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E91">A629</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> holds <br/> for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font> holds <br/>( <font color="Olive" title="b2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b2">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c25">j</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font> &amp; <font color="Maroon" title="c25">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> implies  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_1"/><span class="lab"><font color="Green" title="E92">A630</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_3_21_1"/><span class="lab"><font color="Green" title="E93">A631</font></span>: 
<font color="Maroon" title="c25">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c26">w</font> = <font color="Maroon" title="c25">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_1"><span class="lab"><font color="Green" title="E92">A630</font></span></a>, <a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c27">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font> implies ( <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> ) )</span><br/><span class="kw">assume </span><a NAME="E4:2_1_1_2_7_1_1_1_1_3_21_1"/><span class="lab"><font color="Green" title="E94">A632</font></span>: 
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_1_1"/>
( <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or <font color="Maroon" title="c25">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_1_1_1"/><span class="lab"><font color="Green" title="E95">A633</font></span>: 
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> )</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_21_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_21_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A634</font></span>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_1_1_1_3_21_1_1_1"/>
<font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_21_1"><span class="lab"><font color="Green" title="E94">A632</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_1_1_1"><span class="lab"><font color="Green" title="E95">A633</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> )</span><br/><span class="kw">thus </span><a NAME="E6:2_1_1_2_7_1_1_1_1_3_21_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E56:2_1_1_2"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E53:2_1_1_2"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E23:2_1_1_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E62">A390</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E64">A392</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E66">A394</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_21_1"><span class="lab"><font color="Green" title="E94">A632</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_1_1_1"><span class="lab"><font color="Green" title="E95">A633</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_21_1_1_1"><span class="lab"><font color="Green" title="E96">A634</font></span></a>, <a class="ref" href="seqm_3.html#D1" onmouseover="rs('seqm_3/D1')" onmouseout="rh()">SEQM_3:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c27">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font></span><br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_21_1_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">l1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:2_1_1_2_7_1_1_1_1_3_21_1_1_1"/>
<font color="Maroon" title="c27">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="txt" href="#E54:2_1_1_2"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E63">A391</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_21_1"><span class="lab"><font color="Green" title="E94">A632</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_1_1_1"><span class="lab"><font color="Green" title="E95">A633</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_21_1_1_1"><span class="lab"><font color="Green" title="E96">A634</font></span></a>, <a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_1_1_2"/>
<font color="Maroon" title="c25">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> )</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_21_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_1"><span class="lab"><font color="Green" title="E92">A630</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_21_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_21_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A635</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_21_1_1_2"/><span class="lab"><font color="Green" title="E96">A636</font></span>: 
<font color="Maroon" title="c26">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21"><span class="lab"><font color="Green" title="E90">A628</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_21_1"><span class="lab"><font color="Green" title="E93">A631</font></span></a>, <a class="ref" href="xreal_1.html#T20" onmouseover="rs('xreal_1/T20')" onmouseout="rh()">XREAL_1:20</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_21_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A637</font></span>: 
<font color="Maroon" title="c26">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_21_1_1_2"><span class="lab"><font color="Green" title="E95">A635</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_21_1_1_2"/>
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_21_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_21_1_1_2"><span class="lab"><font color="Green" title="E95">A635</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_21_1_1_2"><span class="lab"><font color="Green" title="E96">A636</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_1_1_1_3_21_1_1_2"/>
( <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">l1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E73">A406</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_21_1"><span class="lab"><font color="Green" title="E94">A632</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_21_1_1_2"><span class="lab"><font color="Green" title="E97">A637</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E3:2_1_1_2_7_1_1_1_1_3_21"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> or <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_21_2"/><span class="lab"><font color="Green" title="E92">A638</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">X</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_21_2"/><span class="lab"><font color="Green" title="E93">A639</font></span>: 
<font color="Maroon" title="c26">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_21_2"/><span class="lab"><font color="Green" title="E94">A640</font></span>: 
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font>)
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_21_2"/><span class="lab"><font color="Green" title="E95">A641</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_21_2"/><span class="lab"><font color="Green" title="E96">A642</font></span>: 
<font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_21_2"><span class="lab"><font color="Green" title="E93">A639</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_2_1"/>
( <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_2_1_1"/><span class="lab"><font color="Green" title="E97">A643</font></span>: 
<font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_21_2_1_1"/>
<font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_21_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_2_1_1"><span class="lab"><font color="Green" title="E97">A643</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_21_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A644</font></span>: 
<font color="Maroon" title="c27">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_21_2"><span class="lab"><font color="Green" title="E95">A641</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_21_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_21_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_2_1_1"><span class="lab"><font color="Green" title="E97">A643</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_21_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c27">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_21_2"><span class="lab"><font color="Green" title="E94">A640</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_21_2_1_1"><span class="lab"><font color="Green" title="E98">A644</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_21_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c26">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_21_2"><span class="lab"><font color="Green" title="E95">A641</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_2_1_1"><span class="lab"><font color="Green" title="E97">A643</font></span></a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_3_21_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A645</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_21_2"><span class="lab"><font color="Green" title="E92">A638</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_3_21_2_1_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T20" onmouseover="rs('topreal3/T20')" onmouseout="rh()">TOPREAL3:20</a></span>;<br/><span class="kw">hence </span><a NAME="E11:2_1_1_2_7_1_1_1_1_3_21_2_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_21_2_1_1"><span class="lab"><font color="Green" title="E99">A645</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_2_1_2"/><span class="lab"><font color="Green" title="E97">A646</font></span>: 
<font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c28">q1</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">i</font>, <font color="Maroon" title="c29">q2</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_21_2_1_2"/><span class="lab"><font color="Green" title="E98">A647</font></span>: 
<font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_21_2"><span class="lab"><font color="Green" title="E96">A642</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_21_2_1_2"/><span class="lab"><font color="Green" title="E99">A648</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E97">A646</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_21_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A649</font></span>: 
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_21"><span class="lab"><font color="Green" title="E91">A629</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E98">A647</font></span></a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_21_2_1_2"/><span class="lab"><font color="Green" title="E101">A650</font></span>: 
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_21"><span class="lab"><font color="Green" title="E91">A629</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E99">A648</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E98">A647</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_21_2_1_2"/><span class="lab"><font color="Green" title="E102">A651</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_21"><span class="lab"><font color="Green" title="E91">A629</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E99">A648</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E98">A647</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_21_2_1_2"/>
<font color="Maroon" title="c29">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_21"><span class="lab"><font color="Green" title="E91">A629</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_21_2"><span class="lab"><font color="Green" title="E96">A642</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E97">A646</font></span></a></span>;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_3_21_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A652</font></span>: 
<font color="Maroon" title="c29">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c29">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E100">A649</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_3_21_2_1_2"/><span class="lab"><font color="Green" title="E104">A653</font></span>: 
<font color="Maroon" title="c29">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_21"><span class="lab"><font color="Green" title="E91">A629</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_21_2"><span class="lab"><font color="Green" title="E96">A642</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E97">A646</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_3_21_2_1_2"/><span class="lab"><font color="Green" title="E105">A654</font></span>: 
( <font color="Maroon" title="c28">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp;  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c29">q2</font>,<font color="Maroon" title="c28">q1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_21_2"><span class="lab"><font color="Green" title="E95">A641</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_21_2"><span class="lab"><font color="Green" title="E96">A642</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_1_1_1_3_21_2_1_2"/><span class="lab"><font color="Green" title="E106">A655</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_21"><span class="lab"><font color="Green" title="E91">A629</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_21_2"><span class="lab"><font color="Green" title="E96">A642</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E97">A646</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_2_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1"/>
( <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  or <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  or <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E103">A652</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E105">A654</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c30">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E107">A656</font></span>: 
( <font color="Maroon" title="c30">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E108">A657</font></span>: 
( <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_21_2"><span class="lab"><font color="Green" title="E92">A638</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_21_2"><span class="lab"><font color="Green" title="E94">A640</font></span></a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E101">A650</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E106">A655</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E108">A657</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A658</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E83">A478</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E100">A649</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E107">A656</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E109">A658</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_2"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c28">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E103">A652</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E105">A654</font></span></a>, <a class="ref" href="rltopsp1.html#T70" onmouseover="rs('rltopsp1/T70')" onmouseout="rh()">RLTOPSP1:70</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_21_2"><span class="lab"><font color="Green" title="E92">A638</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_21_2"><span class="lab"><font color="Green" title="E94">A640</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A659</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E83">A478</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E100">A649</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E102">A651</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E101">A650</font></span></a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E107">A659</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p1</font> where <font color="Olive" title="b1">p1</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E103">A652</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E105">A654</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c30">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"/><span class="lab"><font color="Green" title="E107">A660</font></span>: 
( <font color="Maroon" title="c30">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"/><span class="lab"><font color="Green" title="E108">A661</font></span>: 
( <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_21_2"><span class="lab"><font color="Green" title="E92">A638</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_21_2"><span class="lab"><font color="Green" title="E94">A640</font></span></a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E102">A651</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E104">A653</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E108">A661</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A662</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E83">A478</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_21_2_1_2"><span class="lab"><font color="Green" title="E100">A649</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E107">A660</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_21_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E109">A662</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E14:2_1_1_2_7_1_1_1_1_3_21_2_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> or <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:2_1_1_2_7_1_1_1_1_3_21"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font></span><br/>

<a NAME="E5:2_1_1_2_7_1_1_1_1_3_21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A663</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E1:2_1_1_2"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="ref" href="goboard2.html#T3" onmouseover="rs('goboard2/T3')" onmouseout="rh()">GOBOARD2:3</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_3_1"/>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> or <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_21"><span class="lab"><font color="Green" title="E92">A663</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_3_1_1"/><span class="lab"><font color="Green" title="E93">A664</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_21_3_1_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#T6" onmouseover="rs('goboard2/T6')" onmouseout="rh()">GOBOARD2:6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_1_1_1_3_21_3_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_3_1_1"><span class="lab"><font color="Green" title="E93">A664</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_3_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">p1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_21_3_1_2"/><span class="lab"><font color="Green" title="E93">A665</font></span>: 
<font color="Maroon" title="c26">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_21_3_1_2"/><span class="lab"><font color="Green" title="E94">A666</font></span>: 
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_21_3_1_2"/><span class="lab"><font color="Green" title="E95">A667</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_21_3_1_2"/><span class="lab"><font color="Green" title="E96">A668</font></span>: 
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E83">A478</font></span></a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; (  for <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) );<br/><div><span class="lab"><font color="Green" title="E97">A669</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_3_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]</span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c27">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><span class="kw">take </span><font color="Maroon" title="c28">n</font> = <font color="Maroon" title="c27">n</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c28">n</font>]</span><br/><span class="kw">thus </span><a NAME="E2:2_1_1_2_7_1_1_1_1_3_21_3_1_2_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c28">n</font>]
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_3_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_3_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">n</font> &amp; <font color="Maroon" title="c28">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21"><span class="lab"><font color="Green" title="E90">A628</font></span></a>, <a class="ref" href="xreal_1.html#T31" onmouseover="rs('xreal_1/T31')" onmouseout="rh()">XREAL_1:31</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">n</font> holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/>

<a NAME="E2:2_1_1_2_7_1_1_1_1_3_21_3_1_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E3:2_1_1_2_7_1_1_1_1_3_21_3_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A670</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c29">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c29">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">n</font> implies <font color="Maroon" title="c29">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E4:2_1_1_2_7_1_1_1_1_3_21_3_1_2_1_1"/>
<font color="Maroon" title="c29">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span><br/>

<span class="kw">then </span><font color="Maroon" title="c29">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_21_3_1_2_1_1"><span class="lab"><font color="Green" title="E98">A670</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_1_1_1_3_21_3_1_2_1_1"/>
<font color="Maroon" title="c29">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E95">A667</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E8:2_1_1_2_7_1_1_1_1_3_21_3_1_2"/><span class="lab"><font color="Green" title="E98">A671</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c27">ma</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:2_1_1_2_7_1_1_1_1_3_21_3_1_2"/><span class="lab"><font color="Green" title="E99">A672</font></span>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c27">ma</font>] &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">ma</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S6" onmouseover="rs('nat_1/S6')" onmouseout="rh()">NAT_1:sch 6</a>(<a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E98">A671</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E97">A669</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c28">ma</font> = <font color="Maroon" title="c27">ma</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_3_1_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1"/>
( <font color="Maroon" title="c28">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> or <font color="Maroon" title="c28">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><span class="lab"><font color="Green" title="E100">A673</font></span>: 
<font color="Maroon" title="c28">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/>
<font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E62">A390</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A674</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21"><span class="lab"><font color="Green" title="E90">A628</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E100">A673</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A675</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">m1</font> = <font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c30">q</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">m1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E9:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><span class="lab"><font color="Green" title="E103">A676</font></span>: 
<font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E100">A673</font></span></a>, <a class="ref" href="xreal_1.html#T43" onmouseover="rs('xreal_1/T43')" onmouseout="rh()">XREAL_1:43</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A677</font></span>: 
<font color="Maroon" title="c30">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_21"><span class="lab"><font color="Green" title="E91">A629</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E102">A675</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><span class="lab"><font color="Green" title="E105">A678</font></span>: 
<font color="Maroon" title="c30">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_21"><span class="lab"><font color="Green" title="E91">A629</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E103">A676</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E102">A675</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c31">lq</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">e</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c30">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">e</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/><a NAME="E12:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><span class="lab"><font color="Green" title="E106">A679</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font>
 ;<br/><a NAME="E13:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><span class="lab"><font color="Green" title="E107">A680</font></span>: 
<font color="Maroon" title="c19">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E101">A674</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A681</font></span>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">ma</font> = 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21"><span class="lab"><font color="Green" title="E90">A628</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E100">A673</font></span></a>, <a class="ref" href="finseq_4.html#T69" onmouseover="rs('finseq_4/T69')" onmouseout="rh()">FINSEQ_4:69</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E107">A680</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E106">A679</font></span></a></span>
;<br/><a NAME="E15:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E99">A672</font></span></a></span>;<br/><a NAME="E16:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A682</font></span>: 
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E96">A668</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E17:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E18:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A683</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E102">A675</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E19:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><span class="lab"><font color="Green" title="E111">A684</font></span>: 
<font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">ma</font>
 ;<br/><a NAME="E20:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c30">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp;  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c30">q</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E100">A673</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E108">A681</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E110">A683</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E21:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">e</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c30">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">e</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c7">j2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E80">A462</font></span></a>, <a class="txt" href="#E33:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E82">A477</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E104">A677</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E105">A678</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a></span>;<br/><a NAME="E22:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A685</font></span>: 
<font color="Maroon" title="c26">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E94">A666</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E109">A682</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E105">A678</font></span></a></span>;<br/><a NAME="E23:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E100">A673</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E110">A683</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E111">A684</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E24:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E93">A665</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E112">A685</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/>
<font color="Maroon" title="c28">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c28">ma</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E99">A672</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A686</font></span>: 
<font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c29">qa</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">ma</font>, <font color="Maroon" title="c30">qa1</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c31">lma</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> ;<br/><a NAME="E5:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/><span class="lab"><font color="Green" title="E101">A687</font></span>: 
<font color="Maroon" title="c30">qa1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/><span class="lab"><font color="Green" title="E102">A688</font></span>: 
<font color="Maroon" title="c29">qa</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E99">A672</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/><span class="lab"><font color="Green" title="E103">A689</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E99">A672</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A690</font></span>: 
<font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_21"><span class="lab"><font color="Green" title="E91">A629</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E100">A686</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E105">A691</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2_1"/>
<font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 ;<br/><a NAME="E3:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E99">A672</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E100">A686</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E103">A689</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E10:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/><span class="lab"><font color="Green" title="E106">A692</font></span>: 
( <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c29">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">qa</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c29">qa</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_1_1_1_3_21"><span class="lab"><font color="Green" title="E91">A629</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E99">A672</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/><span class="lab"><font color="Green" title="E107">A693</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E99">A672</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c29">qa</font>,<font color="Maroon" title="c30">qa1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E100">A686</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E102">A688</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E105">A691</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E104">A690</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E106">A692</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E101">A687</font></span></a>, <a class="ref" href="topreal3.html#T9" onmouseover="rs('topreal3/T9')" onmouseout="rh()">TOPREAL3:9</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
;<br/><a NAME="E13:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A694</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E93">A665</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_1_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E94">A666</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E102">A688</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E105">A691</font></span></a></span>;<br/><a NAME="E14:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E100">A686</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E107">A693</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_1_2_7_1_1_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E108">A694</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:2_1_1_2_7_1_1_1_1_3_21_3_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:2_1_1_2_7_1_1_1_1_3_21"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E50:2_1_1_2_7_1_1_1_1_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E51:2_1_1_2_7_1_1_1_1_3"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2"><span class="lab"><font color="Green" title="E37">A45</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E16:2_1_1_2"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )</span><br/><a NAME="E53:2_1_1_2_7_1_1_1_1_3"/><span class="lab"><font color="Green" title="E90">A695</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/><a NAME="E54:2_1_1_2_7_1_1_1_1_3"/>
<font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E62">A390</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E55:2_1_1_2_7_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A696</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E56:2_1_1_2_7_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A697</font></span>: 
<font color="Maroon" title="c19">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> = 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E90">A695</font></span></a>, <a class="ref" href="finseq_4.html#T69" onmouseover="rs('finseq_4/T69')" onmouseout="rh()">FINSEQ_4:69</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">l</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E68">A396</font></span></a>, <a class="txt" href="#E56:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E92">A697</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E12:2_1_1_2"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_1"><span class="lab"><font color="Green" title="E61">A76</font></span></a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span><br/><span class="kw">thus </span><a NAME="E58:2_1_1_2_7_1_1_1_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="txt" href="#E55:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E91">A696</font></span></a>, <a class="txt" href="#E53:2_1_1_2_7_1_1_1_1_3"><span class="lab"><font color="Green" title="E90">A695</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_1"/>
 ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2"/><span class="lab"><font color="Green" title="E61">A698</font></span>: 
<font color="Maroon" title="c9">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c17">ppi</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>);<br/><span class="kw">set </span><font color="Maroon" title="c18">pj</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>);<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c8">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">j1</font> &amp;  ex <font color="Olive" title="b1">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> ) ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j1</font> &amp; contradiction ) or ( <font color="Maroon" title="c8">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j1</font> &amp;  ex <font color="Olive" title="b1">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> ) ) )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1"/>
( <font color="Maroon" title="c8">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">j1</font> or <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j1</font> or <font color="Maroon" title="c8">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j1</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E62">A699</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">j1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> )</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1"/>
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A700</font></span>: 
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A701</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E7:2_1_1_2"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E45:2_1_1_2"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="txt" href="#E46:2_1_1_2"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E41:2_1_1_2"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1"/>
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A702</font></span>: 
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A703</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="txt" href="#E7:2_1_1_2"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E45:2_1_1_2"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="txt" href="#E46:2_1_1_2"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E41:2_1_1_2"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A704</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="txt" href="#E7:2_1_1_2"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E51:2_1_1_2"><span class="lab"><font color="Green" title="E54">A69</font></span></a>, <a class="txt" href="#E45:2_1_1_2"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="txt" href="#E46:2_1_1_2"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E41:2_1_1_2"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E62">A699</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">A701</font></span></a>, <a class="ref" href="seqm_3.html#D1" onmouseover="rs('seqm_3/D1')" onmouseout="rh()">SEQM_3:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">l</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E62">A699</font></span></a>, <a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">m</font>,<font color="Maroon" title="c9">i2</font>);<br/><span class="kw">set </span><font color="Maroon" title="c20">lk</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E68">A705</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><div><span class="lab"><font color="Green" title="E69">A706</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> holds <br/>( <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">n</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> implies ( <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_1_1"/>
<font color="Maroon" title="c21">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> )</span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A707</font></span>: 
<font color="Maroon" title="c21">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_1"/>
<font color="Maroon" title="c19">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T43" onmouseover="rs('xreal_1/T43')" onmouseout="rh()">XREAL_1:43</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">w</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E70">A707</font></span></a>, <a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_1"/>
( <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font> &amp; <font color="Maroon" title="c8">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="ref" href="xreal_1.html#T43" onmouseover="rs('xreal_1/T43')" onmouseout="rh()">XREAL_1:43</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A708</font></span>: 
<font color="Maroon" title="c22">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E72">A709</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_1"/>
<font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E70">A707</font></span></a>, <a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E72">A709</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E71">A708</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">hence </span><a NAME="E11:2_1_1_2_7_1_2_1_1_1_1"/>
( <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_1_2"><span class="lab"><font color="Green" title="E20">A22</font></span></a>, <a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E70">A710</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> holds <br/> ex <font color="Olive" title="b2">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">p</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> implies  ex <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">n</font>,<font color="Olive" title="b1">p</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_1_2"/>
<font color="Maroon" title="c21">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">n</font>,<font color="Olive" title="b1">p</font>]</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">m</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c21">n</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c23">p</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c22">m</font>,<font color="Maroon" title="c9">i2</font>); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">n</font>,<font color="Maroon" title="c23">p</font>]</span><br/><span class="kw">thus </span><a NAME="E3:2_1_1_2_7_1_2_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c21">n</font>,<font color="Maroon" title="c23">p</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c21">g2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E14:2_1_1_2_7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E71">A711</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_4.html#S1" onmouseover="rs('finseq_4/S1')" onmouseout="rh()">FINSEQ_4:sch 1</a>(<a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E70">A710</font></span></a>);<br/></span><span class="kw">take </span><font color="Maroon" title="c22">g</font> = <font color="Maroon" title="c16">g1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c21">g2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c22">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )</span><br/><a NAME="E15:2_1_1_2_7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E72">A712</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> holds <br/> ex <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font>) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> implies  ex <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k</font>) ) )</span><br/><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_1_3"/><span class="lab"><font color="Green" title="E73">A713</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k</font>) )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">m</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c25">m</font> = <font color="Maroon" title="c24">m</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">m</font>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">m</font>,<font color="Olive" title="b1">k</font>) )</span><br/><span class="kw">take </span><font color="Maroon" title="c26">k</font> = <font color="Maroon" title="c9">i2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">m</font>,<font color="Maroon" title="c26">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">m</font>,<font color="Maroon" title="c26">k</font>) )</span><br/><span class="kw">thus </span><a NAME="E3:2_1_1_2_7_1_2_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">m</font>,<font color="Maroon" title="c26">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">m</font>,<font color="Maroon" title="c26">k</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E73">A713</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E17:2_1_1_2_7_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A714</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:2_1_1_2"><span class="lab"><font color="Green" title="E60">A75</font></span></a>, <a class="ref" href="goboard1.html#T23" onmouseover="rs('goboard1/T23')" onmouseout="rh()">GOBOARD1:23</a></span>;<br/><a NAME="E18:2_1_1_2_7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E74">A715</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E19:2_1_1_2_7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E75">A716</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E7:2_1_1_2"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E43:2_1_1_2"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E41:2_1_1_2"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E63">A700</font></span></a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><div><span class="lab"><font color="Green" title="E76">A717</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> holds <br/>( <font color="Olive" title="b2">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> &amp; <font color="Olive" title="b2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> holds <br/>( <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c24">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p</font> implies ( <font color="Maroon" title="c24">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c24">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c24">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> &amp; <font color="Maroon" title="c24">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_1_4"/><span class="lab"><font color="Green" title="E77">A718</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_1_4"/><span class="lab"><font color="Green" title="E78">A719</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">p</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c24">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c24">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> &amp; <font color="Maroon" title="c24">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c25">n1</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E77">A718</font></span></a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_4"/>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A715</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E77">A718</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A720</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T13" onmouseover="rs('xreal_1/T13')" onmouseout="rh()">XREAL_1:13</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c26">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c9">i2</font>);<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_4"/><span class="lab"><font color="Green" title="E80">A721</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c9">i2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A715</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E77">A718</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_4"/><span class="lab"><font color="Green" title="E81">A722</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A715</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E77">A718</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A723</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E7:2_1_1_2"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E52:2_1_1_2"><span class="lab"><font color="Green" title="E55">A70</font></span></a>, <a class="txt" href="#E43:2_1_1_2"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E41:2_1_1_2"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="ref" href="seqm_3.html#D10" onmouseover="rs('seqm_3/D10')" onmouseout="rh()">SEQM_3:def 10</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_1_4"/>
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E81">A722</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A724</font></span>: 
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c9">i2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E81">A722</font></span></a>, <a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A725</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E45:2_1_1_2"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="txt" href="#E46:2_1_1_2"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E41:2_1_1_2"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E81">A722</font></span></a>, <a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_2_1_1_1_4"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E43:2_1_1_2"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E41:2_1_1_2"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E81">A722</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E83">A724</font></span></a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E13:2_1_1_2_7_1_2_1_1_1_4"/>
( <font color="Maroon" title="c24">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c24">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="txt" href="#E7:2_1_1_2"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E51:2_1_1_2"><span class="lab"><font color="Green" title="E54">A69</font></span></a>, <a class="txt" href="#E45:2_1_1_2"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="txt" href="#E46:2_1_1_2"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E41:2_1_1_2"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A716</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">A701</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A703</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E78">A719</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E81">A722</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E80">A721</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E79">A720</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E82">A723</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E84">A725</font></span></a>, <a class="ref" href="seq_4.html#T137" onmouseover="rs('seq_4/T137')" onmouseout="rh()">SEQ_4:137</a>, <a class="ref" href="xreal_1.html#T43" onmouseover="rs('xreal_1/T43')" onmouseout="rh()">XREAL_1:43</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> &amp; <font color="Maroon" title="c24">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/><span class="kw">thus </span><a NAME="E14:2_1_1_2_7_1_2_1_1_1_4"/>
<font color="Maroon" title="c24">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E78">A719</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E81">A722</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E80">A721</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E83">A724</font></span></a>, <a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><a NAME="E15:2_1_1_2_7_1_2_1_1_1_4"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A715</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E77">A718</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E16:2_1_1_2_7_1_2_1_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c25">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T44" onmouseover="rs('xreal_1/T44')" onmouseout="rh()">XREAL_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E17:2_1_1_2_7_1_2_1_1_1_4"/>
<font color="Maroon" title="c24">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E7:2_1_1_2"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E51:2_1_1_2"><span class="lab"><font color="Green" title="E54">A69</font></span></a>, <a class="txt" href="#E45:2_1_1_2"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="txt" href="#E46:2_1_1_2"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E41:2_1_1_2"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">A701</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E78">A719</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E81">A722</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E80">A721</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_4"><span class="lab"><font color="Green" title="E84">A725</font></span></a>, <a class="ref" href="seqm_3.html#D1" onmouseover="rs('seqm_3/D1')" onmouseout="rh()">SEQM_3:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E21:2_1_1_2_7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E77">A726</font></span>: 
<font color="Maroon" title="c21">g2</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topreal1.html#D5" onmouseover="rs('topreal1/D5')" onmouseout="rh()">TOPREAL1:def 5</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> or  not <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c24">p</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>;<br/>
<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_1_5"/><span class="lab"><font color="Green" title="E78">A727</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<a NAME="E2:2_1_1_2_7_1_2_1_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E3:2_1_1_2_7_1_2_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A728</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A717</font></span></a></span>;<br/>
<a NAME="E4:2_1_1_2_7_1_2_1_1_1_5"/>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_5"><span class="lab"><font color="Green" title="E78">A727</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_2_1_1_1_5"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A717</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_5"><span class="lab"><font color="Green" title="E79">A728</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E78">A729</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_6"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q</font> holds <br/><font color="Olive" title="b4">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> holds <br/><font color="Olive" title="b2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c25">p</font>, <font color="Maroon" title="c26">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">p</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q</font> implies <font color="Maroon" title="c26">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_1_6"/><span class="lab"><font color="Green" title="E79">A730</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_1_6"/><span class="lab"><font color="Green" title="E80">A731</font></span>: 
<font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_6"/><span class="lab"><font color="Green" title="E81">A732</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_6"/><span class="lab"><font color="Green" title="E82">A733</font></span>: 
( <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">p</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_6"/><span class="lab"><font color="Green" title="E83">A734</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E79">A730</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c27">n1</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c28">m1</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">m</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E79">A730</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E80">A731</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c29">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c27">n1</font>,<font color="Maroon" title="c9">i2</font>);<br/><span class="kw">set </span><font color="Maroon" title="c30">pm</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c28">m1</font>,<font color="Maroon" title="c9">i2</font>);<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_6"/><span class="lab"><font color="Green" title="E84">A735</font></span>: 
<font color="Maroon" title="c28">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E81">A732</font></span></a>, <a class="ref" href="xreal_1.html#T15" onmouseover="rs('xreal_1/T15')" onmouseout="rh()">XREAL_1:15</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_6"/>
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E79">A730</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c27">n1</font>,<font color="Maroon" title="c9">i2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E83">A734</font></span></a>, <a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A736</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c27">n1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E83">A734</font></span></a>, <a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_1_6"/><span class="lab"><font color="Green" title="E86">A737</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E80">A731</font></span></a></span>;<br/><a NAME="E12:2_1_1_2_7_1_2_1_1_1_6"/>
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E80">A731</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E13:2_1_1_2_7_1_2_1_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c28">m1</font>,<font color="Maroon" title="c9">i2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E86">A737</font></span></a>, <a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>;<br/><a NAME="E14:2_1_1_2_7_1_2_1_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A738</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c28">m1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E86">A737</font></span></a>, <a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><a NAME="E15:2_1_1_2_7_1_2_1_1_1_6"/>
( <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c27">n1</font>,<font color="Maroon" title="c9">i2</font>) &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c28">m1</font>,<font color="Maroon" title="c9">i2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E79">A730</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E80">A731</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E16:2_1_1_2_7_1_2_1_1_1_6"/>
<font color="Maroon" title="c26">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:2_1_1_2"><span class="lab"><font color="Green" title="E54">A69</font></span></a>, <a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E82">A733</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E83">A734</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E86">A737</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E84">A735</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E85">A736</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1_6"><span class="lab"><font color="Green" title="E87">A738</font></span></a>, <a class="ref" href="seqm_3.html#D1" onmouseover="rs('seqm_3/D1')" onmouseout="rh()">SEQM_3:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E23:2_1_1_2_7_1_2_1_1_1"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> holds <br/> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b2">m</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> implies  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_1_7"/><span class="lab"><font color="Green" title="E79">A739</font></span>: 
<font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_1_7"/><span class="lab"><font color="Green" title="E80">A740</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_7"/><span class="lab"><font color="Green" title="E81">A741</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_7"/><span class="lab"><font color="Green" title="E82">A742</font></span>: 
<font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_7"/><span class="lab"><font color="Green" title="E83">A743</font></span>: 
<font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_7"/><span class="lab"><font color="Green" title="E84">A744</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D7" onmouseover="rs('xboole_0/D7')" onmouseout="rh()">XBOOLE_0:def 7</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c25">p1</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c26">p2</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <font color="Maroon" title="c27">q1</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font>, <font color="Maroon" title="c28">q2</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E8:2_1_1_2_7_1_2_1_1_1_7"/><span class="lab"><font color="Green" title="E85">A745</font></span>: 
( <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A717</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E80">A740</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E81">A741</font></span></a></span>;<br/>
<a NAME="E9:2_1_1_2_7_1_2_1_1_1_7"/>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E10:2_1_1_2_7_1_2_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A746</font></span>: 
<font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A729</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E80">A740</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E81">A741</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c29">lp</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c30">lq</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/>
<a NAME="E11:2_1_1_2_7_1_2_1_1_1_7"/><span class="lab"><font color="Green" title="E87">A747</font></span>: 
( <font color="Maroon" title="c25">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c26">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_2_1_1_1_7"/>
<font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_2_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A748</font></span>: 
<font color="Maroon" title="c28">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A729</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E82">A742</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E83">A743</font></span></a></span>;<br/>
<a NAME="E14:2_1_1_2_7_1_2_1_1_1_7"/><span class="lab"><font color="Green" title="E89">A749</font></span>: 
( <font color="Maroon" title="c27">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c28">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c31">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span>;<br/>
<a NAME="E15:2_1_1_2_7_1_2_1_1_1_7"/><span class="lab"><font color="Green" title="E90">A750</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E84">A744</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E16:2_1_1_2_7_1_2_1_1_1_7"/><span class="lab"><font color="Green" title="E91">A751</font></span>: 
( <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A717</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E82">A742</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E83">A743</font></span></a></span>;<br/>
<a NAME="E17:2_1_1_2_7_1_2_1_1_1_7"/><span class="lab"><font color="Green" title="E92">A752</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E84">A744</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E18:2_1_1_2_7_1_2_1_1_1_7"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E82">A742</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E83">A743</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c28">q2</font>,<font color="Maroon" title="c27">q1</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E88">A748</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E91">A751</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E89">A749</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>
;<br/>
<a NAME="E20:2_1_1_2_7_1_2_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A753</font></span>: 
 ex <font color="Olive" title="b1">tm</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">tm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> &amp; <font color="Olive" title="b1">tm</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">tm</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">tm</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E92">A752</font></span></a></span>;<br/>
<a NAME="E21:2_1_1_2_7_1_2_1_1_1_7"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E80">A740</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E81">A741</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c26">p2</font>,<font color="Maroon" title="c25">p1</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E86">A746</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E85">A745</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E87">A747</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>
;<br/>
<a NAME="E23:2_1_1_2_7_1_2_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A754</font></span>: 
 ex <font color="Olive" title="b1">tn</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">tn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> &amp; <font color="Olive" title="b1">tn</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">tn</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">tn</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E90">A750</font></span></a></span>;<br/>
<a NAME="E24:2_1_1_2_7_1_2_1_1_1_7"/>
<font color="Maroon" title="c27">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A729</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E79">A739</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E81">A741</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E82">A742</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:2_1_1_2_7_1_2_1_1_1_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E94">A754</font></span></a>, <a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1_7"><span class="lab"><font color="Green" title="E93">A753</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E24:2_1_1_2_7_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A755</font></span>: 
<font color="Maroon" title="c21">g2</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#T1" onmouseover="rs('goboard2/T1')" onmouseout="rh()">GOBOARD2:1</a></span>;<br/><a NAME="E25:2_1_1_2_7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E80">A756</font></span>: 
 not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c23">ls</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> ) </span> } </span> ;<br/>
<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_1_8"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_8"/><span class="lab"><font color="Green" title="E81">A757</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c24">X</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_8"/><span class="lab"><font color="Green" title="E82">A758</font></span>: 
<font color="Maroon" title="c24">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_8"/><span class="lab"><font color="Green" title="E83">A759</font></span>: 
<font color="Maroon" title="c24">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c25">m</font>)
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_8"/><span class="lab"><font color="Green" title="E84">A760</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">m</font> &amp; <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_8"><span class="lab"><font color="Green" title="E82">A758</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">q1</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">m</font>, <font color="Maroon" title="c27">q2</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E9:2_1_1_2_7_1_2_1_1_1_8"/><span class="lab"><font color="Green" title="E85">A761</font></span>: 
<font color="Maroon" title="c25">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_8"><span class="lab"><font color="Green" title="E84">A760</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E10:2_1_1_2_7_1_2_1_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A762</font></span>: 
<font color="Maroon" title="c26">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A717</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c28">lq</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/>
<a NAME="E11:2_1_1_2_7_1_2_1_1_1_8"/><span class="lab"><font color="Green" title="E87">A763</font></span>: 
( <font color="Maroon" title="c26">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c27">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_2_1_1_1_8"/><span class="lab"><font color="Green" title="E88">A764</font></span>: 
<font color="Maroon" title="c25">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_8"><span class="lab"><font color="Green" title="E84">A760</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_2_1_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A765</font></span>: 
<font color="Maroon" title="c27">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A717</font></span></a></span>;<br/>
<a NAME="E14:2_1_1_2_7_1_2_1_1_1_8"/>
<font color="Maroon" title="c25">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E15:2_1_1_2_7_1_2_1_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A766</font></span>: 
<font color="Maroon" title="c27">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A729</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_8"><span class="lab"><font color="Green" title="E85">A761</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_1_8"><span class="lab"><font color="Green" title="E88">A764</font></span></a></span>;<br/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c25">m</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c27">q2</font>,<font color="Maroon" title="c26">q1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_8"><span class="lab"><font color="Green" title="E84">A760</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_8"><span class="lab"><font color="Green" title="E86">A762</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_1_8"><span class="lab"><font color="Green" title="E89">A765</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1_8"><span class="lab"><font color="Green" title="E90">A766</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_8"><span class="lab"><font color="Green" title="E87">A763</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>
;<br/>
<a NAME="E17:2_1_1_2_7_1_2_1_1_1_8"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_8"><span class="lab"><font color="Green" title="E81">A757</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_8"><span class="lab"><font color="Green" title="E83">A759</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:2_1_1_2_7_1_2_1_1_1_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A717</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_8"><span class="lab"><font color="Green" title="E85">A761</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E26:2_1_1_2_7_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="txt" href="#E7:2_1_1_2"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E43:2_1_1_2"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E41:2_1_1_2"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E65">A702</font></span></a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><a NAME="E27:2_1_1_2_7_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A767</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="txt" href="#E7:2_1_1_2"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E52:2_1_1_2"><span class="lab"><font color="Green" title="E55">A70</font></span></a>, <a class="txt" href="#E43:2_1_1_2"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E41:2_1_1_2"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E75">A716</font></span></a>, <a class="ref" href="seqm_3.html#D10" onmouseover="rs('seqm_3/D10')" onmouseout="rh()">SEQM_3:def 10</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_9"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">m</font> holds <br/> not <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">m</font> implies  not <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_1_9"/><span class="lab"><font color="Green" title="E82">A768</font></span>: 
( <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> )
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_1_9"/><span class="lab"><font color="Green" title="E83">A769</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c25">n1</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c26">m1</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">m</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_9"><span class="lab"><font color="Green" title="E82">A768</font></span></a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_9"/><span class="lab"><font color="Green" title="E84">A770</font></span>: 
( <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c9">i2</font>) &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c26">m1</font>,<font color="Maroon" title="c9">i2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_9"><span class="lab"><font color="Green" title="E82">A768</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E5:2_1_1_2_7_1_2_1_1_1_9"/><span class="lab"><font color="Green" title="E85">A771</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_9"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">m</font></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_9"><span class="lab"><font color="Green" title="E82">A768</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_9"/><span class="kw">then </span>
<font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_9"><span class="lab"><font color="Green" title="E84">A770</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_9"><span class="lab"><font color="Green" title="E85">A771</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E8:2_1_1_2_7_1_2_1_1_1_9"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_9"><span class="lab"><font color="Green" title="E83">A769</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E29:2_1_1_2_7_1_2_1_1_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font>
 ;<br/><a NAME="E30:2_1_1_2_7_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A772</font></span>: 
<font color="Maroon" title="c21">g2</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T9" onmouseover="rs('partfun2/T9')" onmouseout="rh()">PARTFUN2:9</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">m1</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">l</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><a NAME="E32:2_1_1_2_7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E83">A773</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><span class="lab"><font color="Green" title="E84">A774</font></span>:  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E12:2_1_1_2"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2"><span class="lab"><font color="Green" title="E61">A698</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E67">A704</font></span></a>, <a class="txt" href="#E27:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E81">A767</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A705</font></span></a>, <a class="txt" href="#E32:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E83">A773</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>
;<br/><a NAME="E34:2_1_1_2_7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E85">A775</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font> or <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_1_11"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_11"/><span class="lab"><font color="Green" title="E86">A776</font></span>: 
<font color="Maroon" title="c25">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_11"/><span class="lab"><font color="Green" title="E87">A777</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">n1</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c25">n</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A715</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_11"><span class="lab"><font color="Green" title="E86">A776</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c27">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c26">n1</font>,<font color="Maroon" title="c9">i2</font>);<br/>
<a NAME="E6:2_1_1_2_7_1_2_1_1_1_11"/><span class="lab"><font color="Green" title="E88">A778</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c26">n1</font>,<font color="Maroon" title="c9">i2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A715</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_11"><span class="lab"><font color="Green" title="E86">A776</font></span></a></span>;<br/>
<a NAME="E7:2_1_1_2_7_1_2_1_1_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A779</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c26">n1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A717</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_11"><span class="lab"><font color="Green" title="E86">A776</font></span></a></span>;<br/>
<a NAME="E8:2_1_1_2_7_1_2_1_1_1_11"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c26">n1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c26">n1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A717</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_11"><span class="lab"><font color="Green" title="E86">A776</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_11"><span class="lab"><font color="Green" title="E88">A778</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_2_1_1_1_11"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A774</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_11"><span class="lab"><font color="Green" title="E87">A777</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_11"><span class="lab"><font color="Green" title="E88">A778</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_11"><span class="lab"><font color="Green" title="E89">A779</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E86">A780</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_12"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> holds <br/> for <font color="Olive" title="b2">l1</font>, <font color="Olive" title="b3">l2</font>, <font color="Olive" title="b4">l3</font>, <font color="Olive" title="b5">l4</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">l1</font>,<font color="Olive" title="b3">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">l3</font>,<font color="Olive" title="b5">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">l1</font>,<font color="Olive" title="b3">l2</font>) &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b4">l3</font>,<font color="Olive" title="b5">l4</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> &amp; <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> implies  for <font color="Olive" title="b1">l1</font>, <font color="Olive" title="b2">l2</font>, <font color="Olive" title="b3">l3</font>, <font color="Olive" title="b4">l4</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font>) &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_1_12"/><span class="lab"><font color="Green" title="E87">A781</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_1_12"/><span class="lab"><font color="Green" title="E88">A782</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">l1</font>, <font color="Olive" title="b2">l2</font>, <font color="Olive" title="b3">l3</font>, <font color="Olive" title="b4">l4</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font>) &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c25">m1</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">n</font>, <font color="Maroon" title="c26">m2</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_12"><span class="lab"><font color="Green" title="E87">A781</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_12"><span class="lab"><font color="Green" title="E88">A782</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c27">l1</font>, <font color="Maroon" title="c28">l2</font>, <font color="Maroon" title="c29">l3</font>, <font color="Maroon" title="c30">l4</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">l1</font>,<font color="Maroon" title="c28">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">l3</font>,<font color="Maroon" title="c30">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c27">l1</font>,<font color="Maroon" title="c28">l2</font>) &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c29">l3</font>,<font color="Maroon" title="c30">l4</font>) implies <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c29">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c30">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_12"/><span class="lab"><font color="Green" title="E89">A783</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">l1</font>,<font color="Maroon" title="c28">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_12"/><span class="lab"><font color="Green" title="E90">A784</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">l3</font>,<font color="Maroon" title="c30">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_12"/><span class="lab"><font color="Green" title="E91">A785</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c27">l1</font>,<font color="Maroon" title="c28">l2</font>)
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_12"/><span class="lab"><font color="Green" title="E92">A786</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c29">l3</font>,<font color="Maroon" title="c30">l4</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c29">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c30">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_12"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c26">m2</font>,<font color="Maroon" title="c9">i2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_12"><span class="lab"><font color="Green" title="E88">A782</font></span></a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_1_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A787</font></span>: 
( <font color="Maroon" title="c29">l3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">m2</font> &amp; <font color="Maroon" title="c30">l4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_12"><span class="lab"><font color="Green" title="E90">A784</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_12"><span class="lab"><font color="Green" title="E92">A786</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1_12"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">m1</font>,<font color="Maroon" title="c9">i2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_12"><span class="lab"><font color="Green" title="E87">A781</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_1_12"/><span class="kw">then </span>
( <font color="Maroon" title="c27">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">m1</font> &amp; <font color="Maroon" title="c28">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_12"><span class="lab"><font color="Green" title="E89">A783</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_12"><span class="lab"><font color="Green" title="E91">A785</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c29">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c30">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> = 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_12"><span class="lab"><font color="Green" title="E93">A787</font></span></a>, <a class="ref" href="absvalue.html#T2" onmouseover="rs('absvalue/T2')" onmouseout="rh()">ABSVALUE:2</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_13"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">l1</font>, <font color="Olive" title="b2">l2</font>, <font color="Olive" title="b3">l3</font>, <font color="Olive" title="b4">l4</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font>) &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font>) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font> &amp; 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">l1</font>, <font color="Maroon" title="c25">l2</font>, <font color="Maroon" title="c26">l3</font>, <font color="Maroon" title="c27">l4</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">l1</font>,<font color="Maroon" title="c25">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">l3</font>,<font color="Maroon" title="c27">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c24">l1</font>,<font color="Maroon" title="c25">l2</font>) &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c26">l3</font>,<font color="Maroon" title="c27">l4</font>) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font> &amp; 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font> implies <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_1_13"/><span class="lab"><font color="Green" title="E87">A788</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">l1</font>,<font color="Maroon" title="c25">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_1_13"/><span class="lab"><font color="Green" title="E88">A789</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">l3</font>,<font color="Maroon" title="c27">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_13"/><span class="lab"><font color="Green" title="E89">A790</font></span>: 
<font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c24">l1</font>,<font color="Maroon" title="c25">l2</font>)
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_13"/><span class="lab"><font color="Green" title="E90">A791</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c26">l3</font>,<font color="Maroon" title="c27">l4</font>)
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_13"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_13"/><span class="lab"><font color="Green" title="E91">A792</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c28">m1</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_13"><span class="lab"><font color="Green" title="E91">A792</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_13"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c28">m1</font>,<font color="Maroon" title="c9">i2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_13"><span class="lab"><font color="Green" title="E91">A792</font></span></a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_1_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A793</font></span>: 
( <font color="Maroon" title="c26">l3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">m1</font> &amp; <font color="Maroon" title="c27">l4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_13"><span class="lab"><font color="Green" title="E88">A789</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_13"><span class="lab"><font color="Green" title="E90">A791</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1_13"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_1_13"/><span class="kw">then </span>
( <font color="Maroon" title="c24">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> &amp; <font color="Maroon" title="c25">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E20:2_1_1_2"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_13"><span class="lab"><font color="Green" title="E87">A788</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_13"><span class="lab"><font color="Green" title="E89">A790</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> = 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_13"><span class="lab"><font color="Green" title="E92">A793</font></span></a>, <a class="ref" href="absvalue.html#T2" onmouseover="rs('absvalue/T2')" onmouseout="rh()">ABSVALUE:2</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E37:2_1_1_2_7_1_2_1_1_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> holds <br/> for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font>) &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">i</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">j</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:2_1_1_2"><span class="lab"><font color="Green" title="E40">A48</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E86">A780</font></span></a>, <a class="ref" href="goboard1.html#T24" onmouseover="rs('goboard1/T24')" onmouseout="rh()">GOBOARD1:24</a></span>;<br/><span class="kw">hence </span><a NAME="E38:2_1_1_2_7_1_2_1_1_1"/>
<font color="Maroon" title="c22">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E73">A714</font></span></a>, <a class="ref" href="goboard1.html#D9" onmouseover="rs('goboard1/D9')" onmouseout="rh()">GOBOARD1:def 9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )</span><br/><a NAME="E39:2_1_1_2_7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E87">A794</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E12:2_1_1_2"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2"><span class="lab"><font color="Green" title="E61">A698</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E40:2_1_1_2_7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E88">A795</font></span>: 
 not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_14"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_1_14"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_14"/><span class="lab"><font color="Green" title="E89">A796</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_14"/><span class="lab"><font color="Green" title="E90">A797</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c25">n1</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">n</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A715</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_14"><span class="lab"><font color="Green" title="E89">A796</font></span></a></span>;<br/>
<a NAME="E6:2_1_1_2_7_1_2_1_1_1_14"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c24">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">n1</font>,<font color="Maroon" title="c9">i2</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E69">A706</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A715</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_14"><span class="lab"><font color="Green" title="E89">A796</font></span></a></span>;<br/>
<a NAME="E7:2_1_1_2_7_1_2_1_1_1_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A798</font></span>: 
<font color="Maroon" title="c25">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_14"><span class="lab"><font color="Green" title="E90">A797</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/>
<a NAME="E8:2_1_1_2_7_1_2_1_1_1_14"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E74">A715</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_14"><span class="lab"><font color="Green" title="E89">A796</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_2_1_1_1_14"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_14"><span class="lab"><font color="Green" title="E91">A798</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E41:2_1_1_2_7_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_15"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c24">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_1_15"/><span class="lab"><font color="Green" title="E89">A799</font></span>: 
 not <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:2_1_1_2_7_1_2_1_1_1_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A800</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:2_1_1_2_7_1_2_1_1_1_15"/><span class="lab"><font color="Green" title="E91">A801</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_15"><span class="lab"><font color="Green" title="E89">A799</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_15_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_15_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_1_15_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_15_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_15_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_1_2_7_1_2_1_1_1_15_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E40:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E88">A795</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_15"><span class="lab"><font color="Green" title="E91">A801</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_15"><span class="lab"><font color="Green" title="E90">A800</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_15_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_15_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_15_1_1_2"/><span class="kw">then </span>
(  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E38:2_1_1_2"><span class="lab"><font color="Green" title="E41">A49</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E85">A775</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_15"><span class="lab"><font color="Green" title="E91">A801</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_15"><span class="lab"><font color="Green" title="E90">A800</font></span></a>, <a class="ref" href="goboard2.html#T4" onmouseover="rs('goboard2/T4')" onmouseout="rh()">GOBOARD2:4</a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_2_1_1_1_15_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E88">A795</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_15"><span class="lab"><font color="Green" title="E90">A800</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_2_1_1_1_15"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E42:2_1_1_2_7_1_2_1_1_1"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 ;<br/><span class="kw">hence </span><a NAME="E43:2_1_1_2_7_1_2_1_1_1"/>
<font color="Maroon" title="c22">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:2_1_1_2"><span class="lab"><font color="Green" title="E32">A40</font></span></a>, <a class="txt" href="#E30:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E82">A772</font></span></a>, <a class="ref" href="finseq_3.html#T91" onmouseover="rs('finseq_3/T91')" onmouseout="rh()">FINSEQ_3:91</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )</span><br/><a NAME="E44:2_1_1_2_7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E89">A802</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> holds <br/><span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_16"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> &amp; <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> implies <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_1_16"/><span class="lab"><font color="Green" title="E90">A803</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_1_16"/><span class="lab"><font color="Green" title="E91">A804</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E3:2_1_1_2_7_1_2_1_1_1_16"/><span class="lab"><font color="Green" title="E92">A805</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E90">A803</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E91">A804</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/>
<a NAME="E4:2_1_1_2_7_1_2_1_1_1_16"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E5:2_1_1_2_7_1_2_1_1_1_16"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A774</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E85">A775</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_16"/><span class="lab"><font color="Green" title="E93">A806</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u1</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_16"/><span class="lab"><font color="Green" title="E94">A807</font></span>: 
<font color="Maroon" title="c25">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E9:2_1_1_2_7_1_2_1_1_1_16"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1_16"/>
<font color="Maroon" title="c25">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ;<br/>
<a NAME="E11:2_1_1_2_7_1_2_1_1_1_16"/><span class="lab"><font color="Green" title="E95">A808</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E90">A803</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E91">A804</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_2_1_1_1_16"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_2_1_1_1_16"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A774</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E85">A775</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">u2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E15:2_1_1_2_7_1_2_1_1_1_16"/><span class="lab"><font color="Green" title="E96">A809</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u2</font>
 <span class="kw">and </span><br/><a NAME="E16:2_1_1_2_7_1_2_1_1_1_16"/><span class="lab"><font color="Green" title="E97">A810</font></span>: 
<font color="Maroon" title="c26">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E17:2_1_1_2_7_1_2_1_1_1_16"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E18:2_1_1_2_7_1_2_1_1_1_16"/>
<font color="Maroon" title="c26">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ;<br/>
<a NAME="E19:2_1_1_2_7_1_2_1_1_1_16"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E20:2_1_1_2_7_1_2_1_1_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A811</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c25">u1</font>,<font color="Maroon" title="c26">u2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E91">A804</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E93">A806</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E96">A809</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E21:2_1_1_2_7_1_2_1_1_1_16"/>
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E22:2_1_1_2_7_1_2_1_1_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A812</font></span>: 
<font color="Maroon" title="c26">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A729</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E92">A805</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E95">A808</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E93">A806</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E96">A809</font></span></a></span>;<br/>
<a NAME="E23:2_1_1_2_7_1_2_1_1_1_16"/><span class="lab"><font color="Green" title="E100">A813</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E90">A803</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E91">A804</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/>
<a NAME="E24:2_1_1_2_7_1_2_1_1_1_16"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E25:2_1_1_2_7_1_2_1_1_1_16"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A774</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E85">A775</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E27:2_1_1_2_7_1_2_1_1_1_16"/><span class="lab"><font color="Green" title="E101">A814</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">u</font>
 <span class="kw">and </span><br/><a NAME="E28:2_1_1_2_7_1_2_1_1_1_16"/><span class="lab"><font color="Green" title="E102">A815</font></span>: 
<font color="Maroon" title="c27">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E29:2_1_1_2_7_1_2_1_1_1_16"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E30:2_1_1_2_7_1_2_1_1_1_16"/>
<font color="Maroon" title="c27">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ;<br/>
<a NAME="E31:2_1_1_2_7_1_2_1_1_1_16"/>
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E32:2_1_1_2_7_1_2_1_1_1_16"/><span class="kw">then </span>
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E91">A804</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E33:2_1_1_2_7_1_2_1_1_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A816</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c27">u</font>,<font color="Maroon" title="c25">u1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E90">A803</font></span></a>, <a class="txt" href="#E27:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E101">A814</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E93">A806</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c28">lg</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/>
<a NAME="E34:2_1_1_2_7_1_2_1_1_1_16"/>
<font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E35:2_1_1_2_7_1_2_1_1_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A817</font></span>: 
<font color="Maroon" title="c25">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A729</font></span></a>, <a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E100">A813</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E92">A805</font></span></a>, <a class="txt" href="#E27:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E101">A814</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E93">A806</font></span></a></span>;<br/>
<a NAME="E36:2_1_1_2_7_1_2_1_1_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A818</font></span>: 
<font color="Maroon" title="c25">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E94">A807</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E99">A812</font></span></a></span>;<br/>
<a NAME="E37:2_1_1_2_7_1_2_1_1_1_16"/>
( <font color="Maroon" title="c27">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c26">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<a NAME="E38:2_1_1_2_7_1_2_1_1_1_16"/><span class="kw">then </span>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E101">A814</font></span></a>, <a class="txt" href="#E28:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E102">A815</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E96">A809</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E97">A810</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E99">A812</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E104">A817</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E39:2_1_1_2_7_1_2_1_1_1_16"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E101">A814</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E93">A806</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E96">A809</font></span></a>, <a class="txt" href="#E33:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E103">A816</font></span></a>, <a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E98">A811</font></span></a>, <a class="txt" href="#E36:2_1_1_2_7_1_2_1_1_1_16"><span class="lab"><font color="Green" title="E105">A818</font></span></a>, <a class="ref" href="topreal1.html#T8" onmouseover="rs('topreal1/T8')" onmouseout="rh()">TOPREAL1:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E45:2_1_1_2_7_1_2_1_1_1"/>
<font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_17"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topreal1.html#D6" onmouseover="rs('topreal1/D6')" onmouseout="rh()">TOPREAL1:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> or  not <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> or <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_1_17"/><span class="lab"><font color="Green" title="E90">A819</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_1_17"/><span class="lab"><font color="Green" title="E91">A820</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E3:2_1_1_2_7_1_2_1_1_1_17"/><span class="lab"><font color="Green" title="E92">A821</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E91">A820</font></span></a></span>;<br/>
<a NAME="E4:2_1_1_2_7_1_2_1_1_1_17"/>
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E5:2_1_1_2_7_1_2_1_1_1_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A822</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E91">A820</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E6:2_1_1_2_7_1_2_1_1_1_17"/><span class="lab"><font color="Green" title="E94">A823</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<a NAME="E7:2_1_1_2_7_1_2_1_1_1_17"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2
 
;<br/>
<a NAME="E8:2_1_1_2_7_1_2_1_1_1_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A824</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E91">A820</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E94">A823</font></span></a>, <a class="ref" href="xreal_1.html#T20" onmouseover="rs('xreal_1/T20')" onmouseout="rh()">XREAL_1:20</a></span>;<br/>
<a NAME="E9:2_1_1_2_7_1_2_1_1_1_17"/><span class="lab"><font color="Green" title="E96">A825</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E10:2_1_1_2_7_1_2_1_1_1_17"/><span class="lab"><font color="Green" title="E97">A826</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E11:2_1_1_2_7_1_2_1_1_1_17"/><span class="lab"><font color="Green" title="E98">A827</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_17_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_1_17_1"/>
( <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_17_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_17_1_1"/><span class="lab"><font color="Green" title="E99">A828</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_17_1_1"/><span class="lab"><font color="Green" title="E100">A829</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_17_1_1"/><span class="lab"><font color="Green" title="E101">A830</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E90">A819</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E99">A828</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A831</font></span>: 
<font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_17_1_1"/>
<font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E90">A819</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E99">A828</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A832</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c24">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E101">A830</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_17_1_1"/>
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E90">A819</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E99">A828</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_17_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E101">A830</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E100">A829</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_2_1_1_1_17_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_1_2"><span class="lab"><font color="Green" title="E33">A41</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E90">A819</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E99">A828</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E103">A832</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_17_1_1"><span class="lab"><font color="Green" title="E102">A831</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_17_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_17_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_17_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_17_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A833</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><span class="kw">thus </span><a NAME="E4:2_1_1_2_7_1_2_1_1_1_17_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_17_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_17_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/><span class="lab"><font color="Green" title="E100">A834</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E92">A821</font></span></a>, <a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E94">A823</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A835</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A774</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E85">A775</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/><span class="lab"><font color="Green" title="E102">A836</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u1</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/>
<font color="Maroon" title="c25">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E9:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/>
<font color="Maroon" title="c25">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/>
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T68" onmouseover="rs('rltopsp1/T68')" onmouseout="rh()">RLTOPSP1:68</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A837</font></span>: 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<font color="Maroon" title="c25">u1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E87">A794</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E85">A775</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A835</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A836</font></span></a>, <a class="ref" href="topreal1.html#T6" onmouseover="rs('topreal1/T6')" onmouseout="rh()">TOPREAL1:6</a></span>;<br/><a NAME="E13:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E14:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A838</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A834</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A839</font></span>: <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A834</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><a NAME="E1:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E90">A819</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A834</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E2:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A840</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">assume </span><a NAME="E3:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E4:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E106">A840</font></span></a>, <a class="ref" href="topreal1.html#T23" onmouseover="rs('topreal1/T23')" onmouseout="rh()">TOPREAL1:23</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E17:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E18:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A841</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="goboard2.html#T4" onmouseover="rs('goboard2/T4')" onmouseout="rh()">GOBOARD2:4</a></span>;<br/><a NAME="E19:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/><span class="lab"><font color="Green" title="E107">A842</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c24">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="ref" href="topreal3.html#T19" onmouseover="rs('topreal3/T19')" onmouseout="rh()">TOPREAL3:19</a></span>;<br/><a NAME="E20:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/>
<font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E90">A819</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E97">A826</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A834</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E21:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A843</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c24">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E104">A838</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E22:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/>
( <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>) &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E90">A819</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E91">A820</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E96">A825</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E93">A822</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E98">A827</font></span></a>, <a class="ref" href="topreal1.html#T21" onmouseover="rs('topreal1/T21')" onmouseout="rh()">TOPREAL1:21</a></span>;<br/><a NAME="E23:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E24:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A844</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/><a NAME="E25:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E91">A820</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E98">A827</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E94">A823</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A834</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E26:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E98">A827</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A834</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><a NAME="E27:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<font color="Maroon" title="c25">u1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E91">A820</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E96">A825</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E98">A827</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A839</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A836</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E28:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E107">A842</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E106">A841</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E108">A843</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A839</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E103">A837</font></span></a>, <a class="ref" href="xboole_1.html#T27" onmouseover="rs('xboole_1/T27')" onmouseout="rh()">XBOOLE_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E29:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_1"><span class="lab"><font color="Green" title="E109">A844</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_17_1_2"><span class="lab"><font color="Green" title="E99">A833</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A845</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">n1</font> = <font color="Maroon" title="c24">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><span class="kw">thus </span><a NAME="E5:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">n</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E101">A846</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A847</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E91">A820</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E94">A823</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A848</font></span>: 
<font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A846</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A847</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A849</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A774</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E85">A775</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E105">A850</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u1</font>
 <span class="kw">and </span><br/><a NAME="E11:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E106">A851</font></span>: 
<font color="Maroon" title="c26">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E12:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E13:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E107">A852</font></span>: 
<font color="Maroon" title="c26">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ;<br/><a NAME="E14:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E15:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A846</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/><a NAME="E17:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A853</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<font color="Maroon" title="c26">u1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E90">A819</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E93">A822</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E103">A848</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A850</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E18:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E109">A854</font></span>: 
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A847</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E19:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/><a NAME="E20:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A774</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E85">A775</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">u2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E22:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E110">A855</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">u2</font>
 <span class="kw">and </span><br/><a NAME="E23:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E111">A856</font></span>: 
<font color="Maroon" title="c27">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E24:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E25:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E112">A857</font></span>: 
<font color="Maroon" title="c27">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ;<br/><span class="kw">set </span><font color="Maroon" title="c28">lg</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c27">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/><a NAME="E26:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c27">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><a NAME="E27:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A858</font></span>: 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c27">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A705</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E110">A855</font></span></a>, <a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E111">A856</font></span></a>, <a class="txt" href="#E25:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E112">A857</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>;<br/><a NAME="E28:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c27">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A729</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E104">A849</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E109">A854</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A850</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E110">A855</font></span></a></span>;<br/><a NAME="E29:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A859</font></span>: 
<font color="Maroon" title="c26">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c27">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E106">A851</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E107">A852</font></span></a></span>;<br/><a NAME="E30:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A846</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A847</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><a NAME="E31:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c26">u1</font>,<font color="Maroon" title="c27">u2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E91">A820</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E96">A825</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E98">A827</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E103">A848</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A850</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E110">A855</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E32:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E103">A848</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A850</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E110">A855</font></span></a>, <a class="txt" href="#E29:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E114">A859</font></span></a>, <a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E108">A853</font></span></a>, <a class="txt" href="#E27:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E113">A858</font></span></a>, <a class="ref" href="topreal1.html#T8" onmouseover="rs('topreal1/T8')" onmouseout="rh()">TOPREAL1:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A860</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2"><span class="lab"><font color="Green" title="E100">A845</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A861</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c25">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">n</font>
 ;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A862</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c25">n1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E93">A822</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E101">A860</font></span></a>, <a class="ref" href="goboard2.html#T5" onmouseover="rs('goboard2/T5')" onmouseout="rh()">GOBOARD2:5</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E104">A863</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c25">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E93">A822</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E94">A823</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A864</font></span>: 
<font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E104">A863</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E97">A826</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E101">A860</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c25">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E92">A821</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E104">A863</font></span></a>, <a class="ref" href="goboard2.html#T5" onmouseover="rs('goboard2/T5')" onmouseout="rh()">GOBOARD2:5</a></span>;<br/><span class="kw">hence </span><a NAME="E13:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c24">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E89">A802</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1_17"><span class="lab"><font color="Green" title="E95">A824</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E103">A862</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E105">A864</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_17_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E102">A861</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E46:2_1_1_2_7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E90">A865</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_18"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font> or <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c25">ls</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> ) </span> } </span> ;<br/>
<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_1_18"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_18"/><span class="lab"><font color="Green" title="E91">A866</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">X</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_18"/><span class="lab"><font color="Green" title="E92">A867</font></span>: 
<font color="Maroon" title="c26">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_18"/><span class="lab"><font color="Green" title="E93">A868</font></span>: 
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c27">m</font>)
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_18"/><span class="lab"><font color="Green" title="E94">A869</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">m</font> &amp; <font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_18"><span class="lab"><font color="Green" title="E92">A867</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c28">q1</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">m</font>, <font color="Maroon" title="c29">q2</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E9:2_1_1_2_7_1_2_1_1_1_18"/><span class="lab"><font color="Green" title="E95">A870</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c27">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c28">q1</font>,<font color="Maroon" title="c29">q2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_18"><span class="lab"><font color="Green" title="E94">A869</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E10:2_1_1_2_7_1_2_1_1_1_18"/>
<font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_18"><span class="lab"><font color="Green" title="E94">A869</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E11:2_1_1_2_7_1_2_1_1_1_18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A871</font></span>: 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_2_1_1_1_18"/>
<font color="Maroon" title="c27">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_18"><span class="lab"><font color="Green" title="E94">A869</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_2_1_1_1_18"/><span class="kw">then </span>
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E14:2_1_1_2_7_1_2_1_1_1_18"/><span class="kw">then </span>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c28">q1</font>,<font color="Maroon" title="c29">q2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E87">A794</font></span></a>, <a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E85">A775</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_18"><span class="lab"><font color="Green" title="E96">A871</font></span></a>, <a class="ref" href="topreal1.html#T6" onmouseover="rs('topreal1/T6')" onmouseout="rh()">TOPREAL1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:2_1_1_2_7_1_2_1_1_1_18"/>
<font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E87">A794</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_18"><span class="lab"><font color="Green" title="E91">A866</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_18"><span class="lab"><font color="Green" title="E93">A868</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_18"><span class="lab"><font color="Green" title="E95">A870</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E47:2_1_1_2_7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E91">A872</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_19"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_19_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_1_19_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_19_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_19_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_1_2_7_1_2_1_1_1_19_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2"><span class="lab"><font color="Green" title="E43">A52</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_19_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_19_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_19_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_19_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="ref" href="goboard2.html#T4" onmouseover="rs('goboard2/T4')" onmouseout="rh()">GOBOARD2:4</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_19_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A873</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E90">A865</font></span></a>, <a class="ref" href="xboole_1.html#T26" onmouseover="rs('xboole_1/T26')" onmouseout="rh()">XBOOLE_1:26</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_19_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c24">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span>;<br/><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_1_19_1_2_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_1_19_1_2_1"/><span class="kw">then </span>
(  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_19_1_2"><span class="lab"><font color="Green" title="E92">A873</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_2_1_1_1_19_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E80">A756</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_2_1_1_1_19_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E48:2_1_1_2_7_1_2_1_1_1"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> &amp; <font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> holds <br/> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Olive" title="b1">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Olive" title="b2">m</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E92">A874</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E2:2_1_1_2_7_1_2_1_1_1_20"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E93">A875</font></span>: <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> = 
<font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">qq</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">l1</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c26">l2</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> ) </span> } </span> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c27">n</font>, <font color="Maroon" title="c28">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c28">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c27">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> &amp; <font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> &amp; <font color="Maroon" title="c28">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> &amp; <font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font> implies  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E94">A876</font></span>: 
<font color="Maroon" title="c28">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E95">A877</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E96">A878</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E97">A879</font></span>: 
<font color="Maroon" title="c28">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">and </span><br/><a NAME="E9:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E98">A880</font></span>: 
<font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c22">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/>

<a NAME="E10:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E99">A881</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E95">A877</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E11:2_1_1_2_7_1_2_1_1_1_20"/>
<font color="Maroon" title="c6">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E62">A699</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_2_1_1_1_20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A882</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_2_1_1_1_20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A883</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E14:2_1_1_2_7_1_2_1_1_1_20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A884</font></span>: 
( <font color="Maroon" title="c24">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c24">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A717</font></span></a></span>;<br/>
<a NAME="E15:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E103">A885</font></span>: 
<font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">qq</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E100">A882</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/>
<a NAME="E16:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E104">A886</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A717</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E101">A883</font></span></a></span>;<br/>
<a NAME="E17:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E105">A887</font></span>: 
<font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E98">A880</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E18:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E106">A888</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E98">A880</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E19:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E107">A889</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E96">A878</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E20:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E108">A890</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E96">A878</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E21:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E109">A891</font></span>: 
<font color="Maroon" title="c24">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c24">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<a NAME="E22:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E110">A892</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E97">A879</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c29">ql</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c24">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/>
<a NAME="E23:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E111">A893</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E24:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E112">A894</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<a NAME="E25:2_1_1_2_7_1_2_1_1_1_20"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E100">A882</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E113">A895</font></span>:  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c24">qq</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E92">A874</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E93">A875</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E103">A885</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c24">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E68">A705</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E102">A884</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E109">A891</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>
;<br/>
<a NAME="E27:2_1_1_2_7_1_2_1_1_1_20"/><span class="lab"><font color="Green" title="E114">A896</font></span>: 
<font color="Maroon" title="c28">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E28:2_1_1_2_7_1_2_1_1_1_20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A897</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E94">A876</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_1_20_3_1"/>
( <font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_20_3_1_1"/><span class="lab"><font color="Green" title="E116">A898</font></span>: 
<font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_20_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c28">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E114">A896</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_20_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A899</font></span>: 
<font color="Maroon" title="c28">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E110">A892</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_20_3_1_1"/>
<font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E106">A888</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_20_3_1_1"><span class="lab"><font color="Green" title="E116">A898</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_20_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A900</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c28">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_20_3_1_1"><span class="lab"><font color="Green" title="E117">A899</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_20_3_1_1"/><span class="lab"><font color="Green" title="E119">A901</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E115">A897</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_20_3_1_1"><span class="lab"><font color="Green" title="E116">A898</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_20_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E111">A893</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_20_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A902</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E99">A881</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_1_20_3_1_1"/>
<font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E107">A889</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_20_3_1_1"><span class="lab"><font color="Green" title="E119">A901</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1_20_3_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c27">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1_20_3_1_1"><span class="lab"><font color="Green" title="E120">A902</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><span class="kw">hence </span><a NAME="E11:2_1_1_2_7_1_2_1_1_1_20_3_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:2_1_1_2"><span class="lab"><font color="Green" title="E34">A42</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E94">A876</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_20_3_1_1"><span class="lab"><font color="Green" title="E118">A900</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_20_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A903</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">m1</font> = <font color="Maroon" title="c28">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1"/>
( <font color="Maroon" title="c28">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or <font color="Maroon" title="c28">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E117">A904</font></span>: 
<font color="Maroon" title="c28">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E118">A905</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c31">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>) or <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_1"/>
<font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c32">px</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E119">A906</font></span>: 
( <font color="Maroon" title="c32">px</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">x</font> &amp; <font color="Maroon" title="c32">px</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E120">A907</font></span>: 
<font color="Maroon" title="c24">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">px</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E121">A908</font></span>: 
<font color="Maroon" title="c32">px</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E113">A895</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E117">A904</font></span></a></span>;<br/>
<a NAME="E6:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">px</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E104">A886</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E120">A907</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_1"/>
<font color="Maroon" title="c31">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A774</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E119">A906</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E121">A908</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"/>
<font color="Maroon" title="c27">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E94">A876</font></span></a>, <a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E111">A893</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E117">A904</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A909</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E99">A881</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><a NAME="E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E94">A876</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E107">A889</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E117">A904</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E2:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A910</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">assume </span><a NAME="E3:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E120">A910</font></span></a>, <a class="ref" href="topreal1.html#T23" onmouseover="rs('topreal1/T23')" onmouseout="rh()">TOPREAL1:23</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A911</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="goboard2.html#T4" onmouseover="rs('goboard2/T4')" onmouseout="rh()">GOBOARD2:4</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E121">A912</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E94">A876</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E107">A889</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E117">A904</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A913</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c27">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E119">A909</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E94">A876</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E99">A881</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E117">A904</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A914</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E120">A911</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E118">A905</font></span></a>, <a class="ref" href="xboole_1.html#T27" onmouseover="rs('xboole_1/T27')" onmouseout="rh()">XBOOLE_1:27</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c31">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span>;<br/><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_3"/><span class="lab"><font color="Green" title="E124">A915</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A916</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_3"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E123">A914</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E124">A915</font></span></a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A917</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_3"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>;<br/><span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:2_1_1_2"><span class="lab"><font color="Green" title="E32">A40</font></span></a>, <a class="txt" href="#E30:2_1_1_2"><span class="lab"><font color="Green" title="E33">A41</font></span></a>, <a class="txt" href="#E31:2_1_1_2"><span class="lab"><font color="Green" title="E34">A42</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E94">A876</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E117">A904</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E119">A909</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E121">A912</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E122">A913</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E125">A916</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E126">A917</font></span></a>, <a class="ref" href="goboard2.html#T2" onmouseover="rs('goboard2/T2')" onmouseout="rh()">GOBOARD2:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E14:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"/>
<font color="Maroon" title="c28">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A918</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_20_3_1_2"><span class="lab"><font color="Green" title="E116">A903</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A919</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"/>
<font color="Maroon" title="c28">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A920</font></span>: 
<font color="Maroon" title="c30">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E105">A887</font></span></a>, <a class="txt" href="#E24:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E112">A894</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"/>
<font color="Maroon" title="c28">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A921</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c30">m1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E105">A887</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E117">A918</font></span></a>, <a class="ref" href="goboard2.html#T5" onmouseover="rs('goboard2/T5')" onmouseout="rh()">GOBOARD2:5</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A919</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E119">A920</font></span></a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A922</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E122">A923</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E111">A893</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E123">A924</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E99">A881</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E107">A889</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E122">A923</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c27">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E123">A924</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E99">A881</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E122">A923</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E91">A872</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E121">A922</font></span></a>, <a class="ref" href="xboole_1.html#T3" onmouseover="rs('xboole_1/T3')" onmouseout="rh()">XBOOLE_1:3</a>, <a class="ref" href="xboole_1.html#T27" onmouseover="rs('xboole_1/T27')" onmouseout="rh()">XBOOLE_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A925</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">n1</font> = <font color="Maroon" title="c27">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E123">A926</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E124">A927</font></span>: 
<font color="Maroon" title="c27">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">n</font> or <font color="Maroon" title="c27">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E125">A928</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c32">q1</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c30">m1</font>, <font color="Maroon" title="c33">q2</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c34">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c35">q1l</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">v</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/><a NAME="E2:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E126">A929</font></span>: 
( <font color="Maroon" title="c32">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c32">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c33">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c33">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><span class="kw">assume </span><a NAME="E3:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E127">A930</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A931</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E127">A930</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A932</font></span>: 
 ex <font color="Olive" title="b1">qx</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">qx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> &amp; <font color="Olive" title="b1">qx</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c24">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qx</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">qx</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E113">A895</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E125">A928</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E130">A933</font></span>: 
<font color="Maroon" title="c30">m1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A919</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E119">A920</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A934</font></span>: 
<font color="Maroon" title="c32">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A717</font></span></a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E132">A935</font></span>: 
<font color="Maroon" title="c30">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A919</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E119">A920</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A936</font></span>: 
<font color="Maroon" title="c33">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A717</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c30">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A937</font></span>: 
<font color="Maroon" title="c33">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A729</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E130">A933</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E132">A935</font></span></a></span>;<br/> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c30">m1</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c33">q2</font>,<font color="Maroon" title="c32">q1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A919</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E119">A920</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">v</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E131">A934</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E133">A936</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E134">A937</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E126">A929</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>
;<br/><a NAME="E14:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E135">A938</font></span>: 
 ex <font color="Olive" title="b1">qy</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">qy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span>)</span> &amp; <font color="Olive" title="b1">qy</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c33">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qy</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">qy</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A921</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E128">A931</font></span></a></span>;<br/><a NAME="E15:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c30">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c31">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c31">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E94">A876</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E123">A926</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E16:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c30">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E17:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c32">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E78">A729</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E101">A883</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E130">A933</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E18:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E129">A932</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E135">A938</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c27">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E122">A925</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A939</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g2</font>,<font color="Maroon" title="c31">n1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E108">A890</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E124">A927</font></span></a>, <a class="ref" href="goboard2.html#T5" onmouseover="rs('goboard2/T5')" onmouseout="rh()">GOBOARD2:5</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c30">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c31">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_20"><span class="lab"><font color="Green" title="E94">A876</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E123">A926</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E79">A755</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A921</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E125">A939</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E12:2_1_1_2_7_1_2_1_1_1_20_3_1_2_1_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_2_1_1_1_20_3_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E30:2_1_1_2_7_1_2_1_1_1_20"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c27">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E49:2_1_1_2_7_1_2_1_1_1"/>
<font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#T1" onmouseover="rs('goboard2/T1')" onmouseout="rh()">GOBOARD2:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_21"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c24">p</font> = <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c25">q</font> = <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1;<br/><a NAME="E1:2_1_1_2_7_1_2_1_1_1_21"/>
<font color="Maroon" title="c6">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E62">A699</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E2:2_1_1_2_7_1_2_1_1_1_21"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_21"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_21"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A717</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_2_1_1_1_21"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E51:2_1_1_2_7_1_2_1_1_1"/>
<font color="Maroon" title="c22">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:2_1_1_2"><span class="lab"><font color="Green" title="E35">A43</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E77">A726</font></span></a>, <a class="ref" href="goboard2.html#T8" onmouseover="rs('goboard2/T8')" onmouseout="rh()">GOBOARD2:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )</span><br/><span class="kw">thus </span><a NAME="E52:2_1_1_2_7_1_2_1_1_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c24">lg</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">lf</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span> } </span> ;<br/>
<a NAME="E1:2_1_1_2_7_1_2_1_1_1_22"/><span class="lab"><font color="Green" title="E92">A940</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E93">A941</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> holds <br/> for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font> holds <br/>( <font color="Olive" title="b2">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c26">j</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">j</font> &amp; <font color="Maroon" title="c26">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> implies  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">j</font> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_1"/><span class="lab"><font color="Green" title="E94">A942</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">j</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_1_22_1"/><span class="lab"><font color="Green" title="E95">A943</font></span>: 
<font color="Maroon" title="c26">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">j</font> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c27">w</font> = <font color="Maroon" title="c26">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_1"><span class="lab"><font color="Green" title="E94">A942</font></span></a>, <a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c28">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c28">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">j</font> implies ( <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> ) )</span><br/><span class="kw">assume </span><a NAME="E4:2_1_1_2_7_1_2_1_1_1_22_1"/><span class="lab"><font color="Green" title="E96">A944</font></span>: 
<font color="Maroon" title="c28">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">j</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_1_1"/>
( <font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or <font color="Maroon" title="c26">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_1_1_1"/><span class="lab"><font color="Green" title="E97">A945</font></span>: 
<font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> )</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_22_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_22_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A946</font></span>: <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_2_1_1_1_22_1_1_1"/>
<font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_22_1"><span class="lab"><font color="Green" title="E96">A944</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_1_1_1"><span class="lab"><font color="Green" title="E97">A945</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> )</span><br/><span class="kw">thus </span><a NAME="E6:2_1_1_2_7_1_2_1_1_1_22_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="txt" href="#E7:2_1_1_2"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E51:2_1_1_2"><span class="lab"><font color="Green" title="E54">A69</font></span></a>, <a class="txt" href="#E45:2_1_1_2"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="txt" href="#E46:2_1_1_2"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E41:2_1_1_2"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E62">A699</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E64">A701</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A703</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_22_1"><span class="lab"><font color="Green" title="E96">A944</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_1_1_1"><span class="lab"><font color="Green" title="E97">A945</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_22_1_1_1"><span class="lab"><font color="Green" title="E98">A946</font></span></a>, <a class="ref" href="seqm_3.html#D1" onmouseover="rs('seqm_3/D1')" onmouseout="rh()">SEQM_3:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c28">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font></span><br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_22_1_1_1"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:2_1_1_2_7_1_2_1_1_1_22_1_1_1"/>
<font color="Maroon" title="c28">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E7:2_1_1_2"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E41:2_1_1_2"><span class="lab"><font color="Green" title="E44">A59</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E63">A700</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_22_1"><span class="lab"><font color="Green" title="E96">A944</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_1_1_1"><span class="lab"><font color="Green" title="E97">A945</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_22_1_1_1"><span class="lab"><font color="Green" title="E98">A946</font></span></a>, <a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_1_1_2"/>
<font color="Maroon" title="c26">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> )</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_22_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_1"><span class="lab"><font color="Green" title="E94">A942</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_22_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_22_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A947</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_22_1_1_2"/><span class="lab"><font color="Green" title="E98">A948</font></span>: 
<font color="Maroon" title="c27">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22"><span class="lab"><font color="Green" title="E92">A940</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_22_1"><span class="lab"><font color="Green" title="E95">A943</font></span></a>, <a class="ref" href="xreal_1.html#T20" onmouseover="rs('xreal_1/T20')" onmouseout="rh()">XREAL_1:20</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_22_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A949</font></span>: 
<font color="Maroon" title="c27">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_22_1_1_2"><span class="lab"><font color="Green" title="E97">A947</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_22_1_1_2"/>
<font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_22_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_22_1_1_2"><span class="lab"><font color="Green" title="E97">A947</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_22_1_1_2"><span class="lab"><font color="Green" title="E98">A948</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_2_1_1_1_22_1_1_2"/>
( <font color="Maroon" title="c28">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c28">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E76">A717</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_22_1"><span class="lab"><font color="Green" title="E96">A944</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_22_1_1_2"><span class="lab"><font color="Green" title="E99">A949</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E3:2_1_1_2_7_1_2_1_1_1_22"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> or <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_22_2"/><span class="lab"><font color="Green" title="E94">A950</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">X</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_22_2"/><span class="lab"><font color="Green" title="E95">A951</font></span>: 
<font color="Maroon" title="c27">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c28">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_22_2"/><span class="lab"><font color="Green" title="E96">A952</font></span>: 
<font color="Maroon" title="c27">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">i</font>)
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_22_2"/><span class="lab"><font color="Green" title="E97">A953</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">i</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_22_2"/><span class="lab"><font color="Green" title="E98">A954</font></span>: 
<font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_22_2"><span class="lab"><font color="Green" title="E95">A951</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_2_1"/>
( <font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or <font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_2_1_1"/><span class="lab"><font color="Green" title="E99">A955</font></span>: 
<font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_22_2_1_1"/>
<font color="Maroon" title="c28">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_22_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c28">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_2_1_1"><span class="lab"><font color="Green" title="E99">A955</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_22_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A956</font></span>: 
<font color="Maroon" title="c28">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_22_2"><span class="lab"><font color="Green" title="E97">A953</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_22_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_22_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_2_1_1"><span class="lab"><font color="Green" title="E99">A955</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_22_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c28">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_22_2"><span class="lab"><font color="Green" title="E96">A952</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_22_2_1_1"><span class="lab"><font color="Green" title="E100">A956</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_22_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_22_2"><span class="lab"><font color="Green" title="E97">A953</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_2_1_1"><span class="lab"><font color="Green" title="E99">A955</font></span></a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_1_22_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A957</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_22_2"><span class="lab"><font color="Green" title="E94">A950</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1_22_2_1_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T20" onmouseover="rs('topreal3/T20')" onmouseout="rh()">TOPREAL3:20</a></span>;<br/><span class="kw">hence </span><a NAME="E11:2_1_1_2_7_1_2_1_1_1_22_2_1_1"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_22_2_1_1"><span class="lab"><font color="Green" title="E101">A957</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_2_1_2"/><span class="lab"><font color="Green" title="E99">A958</font></span>: 
<font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c29">q1</font> = <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">i</font>, <font color="Maroon" title="c30">q2</font> = <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_22_2_1_2"/><span class="lab"><font color="Green" title="E100">A959</font></span>: 
<font color="Maroon" title="c28">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1_22_2"><span class="lab"><font color="Green" title="E98">A954</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_22_2_1_2"/><span class="lab"><font color="Green" title="E101">A960</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E99">A958</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_22_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A961</font></span>: 
<font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_22"><span class="lab"><font color="Green" title="E93">A941</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E100">A959</font></span></a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_22_2_1_2"/><span class="lab"><font color="Green" title="E103">A962</font></span>: 
<font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_22"><span class="lab"><font color="Green" title="E93">A941</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E101">A960</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E100">A959</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_22_2_1_2"/><span class="lab"><font color="Green" title="E104">A963</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_22"><span class="lab"><font color="Green" title="E93">A941</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E101">A960</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E100">A959</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_22_2_1_2"/>
<font color="Maroon" title="c30">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_22"><span class="lab"><font color="Green" title="E93">A941</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1_22_2"><span class="lab"><font color="Green" title="E98">A954</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E99">A958</font></span></a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_1_22_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A964</font></span>: 
<font color="Maroon" title="c30">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E102">A961</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1_22_2_1_2"/><span class="lab"><font color="Green" title="E106">A965</font></span>: 
<font color="Maroon" title="c30">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_22"><span class="lab"><font color="Green" title="E93">A941</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1_22_2"><span class="lab"><font color="Green" title="E98">A954</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E99">A958</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_1_22_2_1_2"/><span class="lab"><font color="Green" title="E107">A966</font></span>: 
( <font color="Maroon" title="c29">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp;  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c30">q2</font>,<font color="Maroon" title="c29">q1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_22_2"><span class="lab"><font color="Green" title="E97">A953</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1_22_2"><span class="lab"><font color="Green" title="E98">A954</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_2_1_1_1_22_2_1_2"/><span class="lab"><font color="Green" title="E108">A967</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_22"><span class="lab"><font color="Green" title="E93">A941</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1_22_2"><span class="lab"><font color="Green" title="E98">A954</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E99">A958</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_2_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1"/>
( <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"/>
<font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E105">A964</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E107">A966</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E109">A968</font></span>: 
( <font color="Maroon" title="c31">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">x</font> &amp; <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E110">A969</font></span>: 
( <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_22_2"><span class="lab"><font color="Green" title="E94">A950</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_22_2"><span class="lab"><font color="Green" title="E96">A952</font></span></a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E103">A962</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E108">A967</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E110">A969</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A970</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A774</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E102">A961</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E109">A968</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E111">A970</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_2"/>
<font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c29">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E105">A964</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E107">A966</font></span></a>, <a class="ref" href="rltopsp1.html#T70" onmouseover="rs('rltopsp1/T70')" onmouseout="rh()">RLTOPSP1:70</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_22_2"><span class="lab"><font color="Green" title="E94">A950</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_22_2"><span class="lab"><font color="Green" title="E96">A952</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A971</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A774</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E102">A961</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E104">A963</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E103">A962</font></span></a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E109">A971</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"/>
<font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c28">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p1</font> where <font color="Olive" title="b1">p1</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E105">A964</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E107">A966</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"/><span class="lab"><font color="Green" title="E109">A972</font></span>: 
( <font color="Maroon" title="c31">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">x</font> &amp; <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"/><span class="lab"><font color="Green" title="E110">A973</font></span>: 
( <font color="Maroon" title="c29">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_22_2"><span class="lab"><font color="Green" title="E94">A950</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_22_2"><span class="lab"><font color="Green" title="E96">A952</font></span></a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c31">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E104">A963</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E106">A965</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E110">A973</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A974</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A774</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_22_2_1_2"><span class="lab"><font color="Green" title="E102">A961</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E109">A972</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_22_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E111">A974</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E14:2_1_1_2_7_1_2_1_1_1_22_2_1_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c26">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> or <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:2_1_1_2_7_1_2_1_1_1_22"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font></span><br/>

<a NAME="E5:2_1_1_2_7_1_2_1_1_1_22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A975</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E1:2_1_1_2"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="ref" href="goboard2.html#T3" onmouseover="rs('goboard2/T3')" onmouseout="rh()">GOBOARD2:3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_3"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_3"/>
( <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> or <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_22"><span class="lab"><font color="Green" title="E94">A975</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_3_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_3_1"/><span class="lab"><font color="Green" title="E95">A976</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_22_3_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#T6" onmouseover="rs('goboard2/T6')" onmouseout="rh()">GOBOARD2:6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_2_1_1_1_22_3_1"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_3_1"><span class="lab"><font color="Green" title="E95">A976</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_3_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_3_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c27">p1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_22_3_2"/><span class="lab"><font color="Green" title="E95">A977</font></span>: 
<font color="Maroon" title="c27">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">x</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_22_3_2"/><span class="lab"><font color="Green" title="E96">A978</font></span>: 
<font color="Maroon" title="c27">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_22_3_2"/><span class="lab"><font color="Green" title="E97">A979</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_22_3_2"/><span class="lab"><font color="Green" title="E98">A980</font></span>: 
<font color="Maroon" title="c27">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A774</font></span></a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> &amp; (  for <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c27">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) );<br/><div><span class="lab"><font color="Green" title="E99">A981</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_3_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]</span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c28">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><span class="kw">take </span><font color="Maroon" title="c29">n</font> = <font color="Maroon" title="c28">n</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c29">n</font>]</span><br/><span class="kw">thus </span><a NAME="E2:2_1_1_2_7_1_2_1_1_1_22_3_2_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c29">n</font>]
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_3_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_3_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">n</font> &amp; <font color="Maroon" title="c29">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22"><span class="lab"><font color="Green" title="E92">A940</font></span></a>, <a class="ref" href="xreal_1.html#T31" onmouseover="rs('xreal_1/T31')" onmouseout="rh()">XREAL_1:31</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">n</font> holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c27">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/>

<a NAME="E2:2_1_1_2_7_1_2_1_1_1_22_3_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E3:2_1_1_2_7_1_2_1_1_1_22_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A982</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c30">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c30">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">n</font> implies <font color="Maroon" title="c30">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c27">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E4:2_1_1_2_7_1_2_1_1_1_22_3_2_1_1"/>
<font color="Maroon" title="c30">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c27">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/>

<span class="kw">then </span><font color="Maroon" title="c30">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_22_3_2_1_1"><span class="lab"><font color="Green" title="E100">A982</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_2_1_1_1_22_3_2_1_1"/>
<font color="Maroon" title="c30">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c27">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E98">A980</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E8:2_1_1_2_7_1_2_1_1_1_22_3_2"/><span class="lab"><font color="Green" title="E100">A983</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c28">ma</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1_22_3_2"/><span class="lab"><font color="Green" title="E101">A984</font></span>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c28">ma</font>] &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S6" onmouseover="rs('nat_1/S6')" onmouseout="rh()">NAT_1:sch 6</a>(<a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E100">A983</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E99">A981</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c29">ma</font> = <font color="Maroon" title="c28">ma</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_3_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_3_2_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1"/>
( <font color="Maroon" title="c29">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> or <font color="Maroon" title="c29">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><span class="lab"><font color="Green" title="E102">A985</font></span>: 
<font color="Maroon" title="c29">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/>
<font color="Maroon" title="c6">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E62">A699</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A986</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22"><span class="lab"><font color="Green" title="E92">A940</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E102">A985</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A987</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">m1</font> = <font color="Maroon" title="c29">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c31">q</font> = <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c30">m1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><span class="lab"><font color="Green" title="E105">A988</font></span>: 
<font color="Maroon" title="c29">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E102">A985</font></span></a>, <a class="ref" href="xreal_1.html#T43" onmouseover="rs('xreal_1/T43')" onmouseout="rh()">XREAL_1:43</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A989</font></span>: 
<font color="Maroon" title="c31">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_22"><span class="lab"><font color="Green" title="E93">A941</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E104">A987</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><span class="lab"><font color="Green" title="E107">A990</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_22"><span class="lab"><font color="Green" title="E93">A941</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E105">A988</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E104">A987</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c32">lq</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">e</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">e</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/><a NAME="E12:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><span class="lab"><font color="Green" title="E108">A991</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j1</font>
 ;<br/><a NAME="E13:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><span class="lab"><font color="Green" title="E109">A992</font></span>: 
<font color="Maroon" title="c19">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E103">A986</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A993</font></span>: <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">ma</font> = 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22"><span class="lab"><font color="Green" title="E92">A940</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E102">A985</font></span></a>, <a class="ref" href="finseq_4.html#T69" onmouseover="rs('finseq_4/T69')" onmouseout="rh()">FINSEQ_4:69</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E109">A992</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E108">A991</font></span></a></span>
;<br/><a NAME="E15:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E101">A984</font></span></a></span>;<br/><a NAME="E16:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A994</font></span>: 
<font color="Maroon" title="c27">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E97">A979</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E17:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E18:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">A995</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E104">A987</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E19:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><span class="lab"><font color="Green" title="E113">A996</font></span>: 
<font color="Maroon" title="c30">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">ma</font>
 ;<br/><a NAME="E20:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c31">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp;  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c30">m1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<font color="Maroon" title="c31">q</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E102">A985</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E110">A993</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E112">A995</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E21:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c30">m1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">e</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">e</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E81">A767</font></span></a>, <a class="txt" href="#E32:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E83">A773</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E106">A989</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E107">A990</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>;<br/><a NAME="E22:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A997</font></span>: 
<font color="Maroon" title="c27">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c30">m1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E96">A978</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E111">A994</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E107">A990</font></span></a></span>;<br/><a NAME="E23:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c30">m1</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E102">A985</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E112">A995</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E113">A996</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E24:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E95">A977</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_1"><span class="lab"><font color="Green" title="E114">A997</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/>
<font color="Maroon" title="c29">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c29">ma</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E101">A984</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A998</font></span>: 
<font color="Maroon" title="c29">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c30">qa</font> = <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">ma</font>, <font color="Maroon" title="c31">qa1</font> = <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c32">lma</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c31">qa1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/><span class="lab"><font color="Green" title="E103">A999</font></span>: 
<font color="Maroon" title="c31">qa1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">qa1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">qa1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/><span class="lab"><font color="Green" title="E104">A1000</font></span>: 
<font color="Maroon" title="c27">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E101">A984</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/><span class="lab"><font color="Green" title="E105">A1001</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E101">A984</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A1002</font></span>: 
<font color="Maroon" title="c31">qa1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_22"><span class="lab"><font color="Green" title="E93">A941</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E102">A998</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E107">A1003</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c27">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">qa1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2_1"/>
<font color="Maroon" title="c27">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">qa1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c27">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c29">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E101">A984</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E102">A998</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E105">A1001</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E10:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/><span class="lab"><font color="Green" title="E108">A1004</font></span>: 
( <font color="Maroon" title="c30">qa</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c30">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">qa</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">qa</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_1_22"><span class="lab"><font color="Green" title="E93">A941</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E101">A984</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/><span class="lab"><font color="Green" title="E109">A1005</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E101">A984</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c29">ma</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c31">qa1</font>,<font color="Maroon" title="c30">qa</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E102">A998</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c31">qa1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E104">A1000</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E107">A1003</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E106">A1002</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E108">A1004</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E103">A999</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
;<br/><a NAME="E13:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A1006</font></span>: 
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c29">ma</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E95">A977</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_1_22_3_2"><span class="lab"><font color="Green" title="E96">A978</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E104">A1000</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E107">A1003</font></span></a></span>;<br/><a NAME="E14:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Maroon" title="c29">ma</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c22">g</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E102">A998</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E109">A1005</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_1_22_3_2_2_1_2"><span class="lab"><font color="Green" title="E110">A1006</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:2_1_1_2_7_1_2_1_1_1_22_3_2"/>
<font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c22">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E53:2_1_1_2_7_1_2_1_1_1"/><span class="lab"><font color="Green" title="E92">A1007</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/><a NAME="E54:2_1_1_2_7_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E55:2_1_1_2_7_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2"><span class="lab"><font color="Green" title="E37">A45</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E16:2_1_1_2"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font> )</span><br/><a NAME="E57:2_1_1_2_7_1_2_1_1_1"/>
<font color="Maroon" title="c6">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E62">A699</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E58:2_1_1_2_7_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A1008</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E59:2_1_1_2_7_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A1009</font></span>: 
<font color="Maroon" title="c19">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c22">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span>)</span> = 
<font color="Maroon" title="c21">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E53:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E92">A1007</font></span></a>, <a class="ref" href="finseq_4.html#T69" onmouseover="rs('finseq_4/T69')" onmouseout="rh()">FINSEQ_4:69</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c23">m1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A712</font></span></a>, <a class="txt" href="#E59:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E94">A1009</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E12:2_1_1_2"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2"><span class="lab"><font color="Green" title="E61">A698</font></span></a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font></span><br/><span class="kw">thus </span><a NAME="E61:2_1_1_2_7_1_2_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c22">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A711</font></span></a>, <a class="txt" href="#E58:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E93">A1008</font></span></a>, <a class="txt" href="#E53:2_1_1_2_7_1_2_1_1_1"><span class="lab"><font color="Green" title="E92">A1007</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_2"/><span class="lab"><font color="Green" title="E62">A1010</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E9:2_1_1_2"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E12:2_1_1_2"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2"><span class="lab"><font color="Green" title="E61">A698</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_2"><span class="lab"><font color="Green" title="E62">A1010</font></span></a>, <a class="ref" href="partfun2.html#T10" onmouseover="rs('partfun2/T10')" onmouseout="rh()">PARTFUN2:10</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3"><span class="kw">case </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3"/><span class="lab"><font color="Green" title="E62">A1011</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> )</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3"/>
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A1012</font></span>: 
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A1013</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3"/>
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A1014</font></span>: 
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A1015</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A1016</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="txt" href="#E51:2_1_1_2"><span class="lab"><font color="Green" title="E54">A69</font></span></a>, <a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E62">A1011</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E64">A1013</font></span></a>, <a class="ref" href="seqm_3.html#D1" onmouseover="rs('seqm_3/D1')" onmouseout="rh()">SEQM_3:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">l</font> = <font color="Maroon" title="c6">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E62">A1011</font></span></a>, <a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1</span>)</span>,<font color="Maroon" title="c9">i2</font>);<br/><span class="kw">consider </span><font color="Maroon" title="c20">g2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E11:2_1_1_2_7_1_2_1_1_3"/><span class="lab"><font color="Green" title="E68">A1017</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">l</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> holds <br/><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_4.html#S2" onmouseover="rs('finseq_4/S2')" onmouseout="rh()">FINSEQ_4:sch 2</a>();<br/></span><span class="kw">take </span><font color="Maroon" title="c21">g</font> = <font color="Maroon" title="c16">g1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c20">g2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )</span><br/><div><span class="lab"><font color="Green" title="E69">A1018</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> holds <br/>( <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> implies ( <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> ) )</span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_3_1"/><span class="lab"><font color="Green" title="E70">A1019</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><span class="kw">assume </span><a NAME="E2:2_1_1_2_7_1_2_1_1_3_1"/><span class="lab"><font color="Green" title="E71">A1020</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )</span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A1021</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_1"/>
<font color="Maroon" title="c6">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A1022</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E72">A1021</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E71">A1020</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E70">A1019</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_2_1_1_3_1"/>
<font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E73">A1022</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font></span><br/><span class="kw">hence </span><a NAME="E10:2_1_1_2_7_1_2_1_1_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_1_2"><span class="lab"><font color="Green" title="E20">A22</font></span></a>, <a class="txt" href="#E50:2_1_1_2"><span class="lab"><font color="Green" title="E53">A68</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E13:2_1_1_2_7_1_2_1_1_3"/><span class="lab"><font color="Green" title="E70">A1023</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> holds <br/> ex <font color="Olive" title="b2">m</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ex <font color="Olive" title="b3">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font>) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> implies  ex <font color="Olive" title="b1">m</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ex <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k</font>) ) )</span><br/><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_3_2"/><span class="lab"><font color="Green" title="E71">A1024</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ex <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">m</font>,<font color="Olive" title="b2">k</font>) )</span><br/><span class="kw">take </span><font color="Maroon" title="c23">m</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">m</font>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c23">m</font>,<font color="Olive" title="b1">k</font>) )</span><br/><span class="kw">take </span><font color="Maroon" title="c24">k</font> = <font color="Maroon" title="c9">i2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c24">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c24">k</font>) )</span><br/><span class="kw">thus </span><a NAME="E2:2_1_1_2_7_1_2_1_1_3_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c24">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c23">m</font>,<font color="Maroon" title="c24">k</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E69">A1018</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E70">A1023</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_2"><span class="lab"><font color="Green" title="E71">A1024</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E15:2_1_1_2_7_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A1025</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:2_1_1_2"><span class="lab"><font color="Green" title="E60">A75</font></span></a>, <a class="ref" href="goboard1.html#T23" onmouseover="rs('goboard1/T23')" onmouseout="rh()">GOBOARD1:23</a></span>;<br/><a NAME="E16:2_1_1_2_7_1_2_1_1_3"/><span class="lab"><font color="Green" title="E72">A1026</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E45:2_1_1_2"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="txt" href="#E46:2_1_1_2"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E43:2_1_1_2"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E63">A1012</font></span></a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><div><span class="lab"><font color="Green" title="E73">A1027</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> holds <br/>( <font color="Olive" title="b2">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> &amp; <font color="Olive" title="b2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> holds <br/>( <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c23">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p</font> implies ( <font color="Maroon" title="c23">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c23">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c23">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> &amp; <font color="Maroon" title="c23">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_3_3"/><span class="lab"><font color="Green" title="E74">A1028</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_3_3"/><span class="lab"><font color="Green" title="E75">A1029</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c23">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c23">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> &amp; <font color="Maroon" title="c23">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_3"/><span class="lab"><font color="Green" title="E76">A1030</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E74">A1028</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c24">n1</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font>;<br/><span class="kw">set </span><font color="Maroon" title="c25">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>);<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_3"/><span class="lab"><font color="Green" title="E77">A1031</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E69">A1018</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E70">A1023</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E74">A1028</font></span></a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A1032</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E52:2_1_1_2"><span class="lab"><font color="Green" title="E55">A70</font></span></a>, <a class="txt" href="#E45:2_1_1_2"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="txt" href="#E46:2_1_1_2"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E43:2_1_1_2"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="ref" href="seqm_3.html#D10" onmouseover="rs('seqm_3/D10')" onmouseout="rh()">SEQM_3:def 10</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_3"/>
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E69">A1018</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E70">A1023</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E74">A1028</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A1033</font></span>: 
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E77">A1031</font></span></a>, <a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A1034</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E77">A1031</font></span></a>, <a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_3_3"/>
<font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E74">A1028</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A1035</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_3_3"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:2_1_1_2"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="txt" href="#E46:2_1_1_2"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E43:2_1_1_2"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E77">A1031</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E79">A1033</font></span></a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E12:2_1_1_2_7_1_2_1_1_3_3"/>
( <font color="Maroon" title="c23">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c23">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="txt" href="#E51:2_1_1_2"><span class="lab"><font color="Green" title="E54">A69</font></span></a>, <a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E72">A1026</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E64">A1013</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E66">A1015</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E75">A1029</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E76">A1030</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E77">A1031</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E81">A1035</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E78">A1032</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E80">A1034</font></span></a>, <a class="ref" href="seq_4.html#T137" onmouseover="rs('seq_4/T137')" onmouseout="rh()">SEQ_4:137</a>, <a class="ref" href="xreal_1.html#T31" onmouseover="rs('xreal_1/T31')" onmouseout="rh()">XREAL_1:31</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> &amp; <font color="Maroon" title="c23">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/><span class="kw">thus </span><a NAME="E13:2_1_1_2_7_1_2_1_1_3_3"/>
<font color="Maroon" title="c23">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E75">A1029</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E76">A1030</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E77">A1031</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E79">A1033</font></span></a>, <a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><a NAME="E14:2_1_1_2_7_1_2_1_1_3_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E74">A1028</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E15:2_1_1_2_7_1_2_1_1_3_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c22">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>;<br/><span class="kw">hence </span><a NAME="E16:2_1_1_2_7_1_2_1_1_3_3"/>
<font color="Maroon" title="c23">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E51:2_1_1_2"><span class="lab"><font color="Green" title="E54">A69</font></span></a>, <a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E64">A1013</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E75">A1029</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E76">A1030</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E77">A1031</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_3"><span class="lab"><font color="Green" title="E80">A1034</font></span></a>, <a class="ref" href="seqm_3.html#D1" onmouseover="rs('seqm_3/D1')" onmouseout="rh()">SEQM_3:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E18:2_1_1_2_7_1_2_1_1_3"/><span class="lab"><font color="Green" title="E74">A1036</font></span>: 
<font color="Maroon" title="c20">g2</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topreal1.html#D5" onmouseover="rs('topreal1/D5')" onmouseout="rh()">TOPREAL1:def 5</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font> or  not <font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c23">p</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font>;<br/>
<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_3_4"/><span class="lab"><font color="Green" title="E75">A1037</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font> &amp; <font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span><br/>

<a NAME="E2:2_1_1_2_7_1_2_1_1_3_4"/><span class="kw">then </span>
<font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E3:2_1_1_2_7_1_2_1_1_3_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A1038</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E73">A1027</font></span></a></span>;<br/>
<a NAME="E4:2_1_1_2_7_1_2_1_1_3_4"/>
<font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_4"><span class="lab"><font color="Green" title="E75">A1037</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_2_1_1_3_4"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E73">A1027</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_4"><span class="lab"><font color="Green" title="E76">A1038</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">m</font> holds <br/> not <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">n</font>, <font color="Maroon" title="c23">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c22">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">m</font> implies  not <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">m</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_3_5"/><span class="lab"><font color="Green" title="E75">A1039</font></span>: 
( <font color="Maroon" title="c22">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> )
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_3_5"/><span class="lab"><font color="Green" title="E76">A1040</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">m</font></span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_5"/><span class="lab"><font color="Green" title="E77">A1041</font></span>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c22">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>) &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">m</font></span>)</span>,<font color="Maroon" title="c9">i2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_5"><span class="lab"><font color="Green" title="E75">A1039</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E4:2_1_1_2_7_1_2_1_1_3_5"/><span class="lab"><font color="Green" title="E78">A1042</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">m</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_5"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c22">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">m</font></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E69">A1018</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E70">A1023</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_5"><span class="lab"><font color="Green" title="E75">A1039</font></span></a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_5"/><span class="kw">then </span>
<font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_5"><span class="lab"><font color="Green" title="E77">A1041</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_5"><span class="lab"><font color="Green" title="E78">A1042</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E7:2_1_1_2_7_1_2_1_1_3_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_5"><span class="lab"><font color="Green" title="E76">A1040</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E20:2_1_1_2_7_1_2_1_1_3"/><span class="kw">then </span>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font>
 ;<br/><a NAME="E21:2_1_1_2_7_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A1043</font></span>: 
<font color="Maroon" title="c20">g2</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T9" onmouseover="rs('partfun2/T9')" onmouseout="rh()">PARTFUN2:9</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c22">lk</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/><a NAME="E22:2_1_1_2_7_1_2_1_1_3"/><span class="lab"><font color="Green" title="E76">A1044</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><div><span class="lab"><font color="Green" title="E77">A1045</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_6"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q</font> holds <br/><font color="Olive" title="b3">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> holds <br/><font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c25">p</font>, <font color="Maroon" title="c26">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">p</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q</font> implies <font color="Maroon" title="c25">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_3_6"/><span class="lab"><font color="Green" title="E78">A1046</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_3_6"/><span class="lab"><font color="Green" title="E79">A1047</font></span>: 
<font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_6"/><span class="lab"><font color="Green" title="E80">A1048</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_6"/><span class="lab"><font color="Green" title="E81">A1049</font></span>: 
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">p</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_6"/><span class="lab"><font color="Green" title="E82">A1050</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E69">A1018</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E70">A1023</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_6"><span class="lab"><font color="Green" title="E78">A1046</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c27">n1</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font>;<br/><span class="kw">set </span><font color="Maroon" title="c28">m1</font> = <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font>;<br/><span class="kw">set </span><font color="Maroon" title="c29">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>);<br/><span class="kw">set </span><font color="Maroon" title="c30">pm</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span>,<font color="Maroon" title="c9">i2</font>);<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_6"/><span class="lab"><font color="Green" title="E83">A1051</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_6"><span class="lab"><font color="Green" title="E80">A1048</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_6"/>
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E69">A1018</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E70">A1023</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_6"><span class="lab"><font color="Green" title="E78">A1046</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_6"/><span class="kw">then </span>
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_6"><span class="lab"><font color="Green" title="E82">A1050</font></span></a>, <a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_3_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A1052</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_6"><span class="lab"><font color="Green" title="E82">A1050</font></span></a>, <a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_3_6"/><span class="lab"><font color="Green" title="E85">A1053</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E69">A1018</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E70">A1023</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_6"><span class="lab"><font color="Green" title="E79">A1047</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_3_6"/>
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E69">A1018</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E70">A1023</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_6"><span class="lab"><font color="Green" title="E79">A1047</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_2_1_1_3_6"/><span class="kw">then </span>
<font color="Maroon" title="c11">c1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span>,<font color="Maroon" title="c9">i2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_6"><span class="lab"><font color="Green" title="E85">A1053</font></span></a>, <a class="ref" href="matrix_0.html#D8" onmouseover="rs('matrix_0/D8')" onmouseout="rh()">MATRIX_0:def 8</a></span>;<br/><a NAME="E13:2_1_1_2_7_1_2_1_1_3_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A1054</font></span>: 
<span class="p1">(<span class="default"><a href="goboard1.html#K1" title="GOBOARD1:func.1">X_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_6"><span class="lab"><font color="Green" title="E85">A1053</font></span></a>, <a class="ref" href="goboard1.html#D1" onmouseover="rs('goboard1/D1')" onmouseout="rh()">GOBOARD1:def 1</a></span>;<br/><a NAME="E14:2_1_1_2_7_1_2_1_1_3_6"/>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>) &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">m</font></span>)</span>,<font color="Maroon" title="c9">i2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_6"><span class="lab"><font color="Green" title="E78">A1046</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_6"><span class="lab"><font color="Green" title="E79">A1047</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:2_1_1_2_7_1_2_1_1_3_6"/>
<font color="Maroon" title="c25">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:2_1_1_2"><span class="lab"><font color="Green" title="E54">A69</font></span></a>, <a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_6"><span class="lab"><font color="Green" title="E81">A1049</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_6"><span class="lab"><font color="Green" title="E82">A1050</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_6"><span class="lab"><font color="Green" title="E85">A1053</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_6"><span class="lab"><font color="Green" title="E83">A1051</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_6"><span class="lab"><font color="Green" title="E84">A1052</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3_6"><span class="lab"><font color="Green" title="E86">A1054</font></span></a>, <a class="ref" href="seqm_3.html#D1" onmouseover="rs('seqm_3/D1')" onmouseout="rh()">SEQM_3:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E24:2_1_1_2_7_1_2_1_1_3"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> holds <br/> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b2">m</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c24">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> implies  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_3_7"/><span class="lab"><font color="Green" title="E78">A1055</font></span>: 
<font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_3_7"/><span class="lab"><font color="Green" title="E79">A1056</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_7"/><span class="lab"><font color="Green" title="E80">A1057</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_7"/><span class="lab"><font color="Green" title="E81">A1058</font></span>: 
<font color="Maroon" title="c24">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_7"/><span class="lab"><font color="Green" title="E82">A1059</font></span>: 
<font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_7"/><span class="lab"><font color="Green" title="E83">A1060</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D7" onmouseover="rs('xboole_0/D7')" onmouseout="rh()">XBOOLE_0:def 7</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c25">p1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font>, <font color="Maroon" title="c26">p2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <font color="Maroon" title="c27">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">m</font>, <font color="Maroon" title="c28">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E8:2_1_1_2_7_1_2_1_1_3_7"/><span class="lab"><font color="Green" title="E84">A1061</font></span>: 
( <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E73">A1027</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E79">A1056</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E80">A1057</font></span></a></span>;<br/>
<a NAME="E9:2_1_1_2_7_1_2_1_1_3_7"/>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E10:2_1_1_2_7_1_2_1_1_3_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A1062</font></span>: 
<font color="Maroon" title="c25">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A1045</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E79">A1056</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E80">A1057</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c29">lp</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c30">lq</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/>
<a NAME="E11:2_1_1_2_7_1_2_1_1_3_7"/><span class="lab"><font color="Green" title="E86">A1063</font></span>: 
( <font color="Maroon" title="c25">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c26">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_2_1_1_3_7"/>
<font color="Maroon" title="c24">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_2_1_1_3_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A1064</font></span>: 
<font color="Maroon" title="c27">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A1045</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E81">A1058</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E82">A1059</font></span></a></span>;<br/>
<a NAME="E14:2_1_1_2_7_1_2_1_1_3_7"/><span class="lab"><font color="Green" title="E88">A1065</font></span>: 
( <font color="Maroon" title="c27">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c28">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c31">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span>;<br/>
<a NAME="E15:2_1_1_2_7_1_2_1_1_3_7"/><span class="lab"><font color="Green" title="E89">A1066</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E83">A1060</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E16:2_1_1_2_7_1_2_1_1_3_7"/><span class="lab"><font color="Green" title="E90">A1067</font></span>: 
( <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E73">A1027</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E81">A1058</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E82">A1059</font></span></a></span>;<br/>
<a NAME="E17:2_1_1_2_7_1_2_1_1_3_7"/><span class="lab"><font color="Green" title="E91">A1068</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E83">A1060</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E18:2_1_1_2_7_1_2_1_1_3_7"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">m</font> &amp; <font color="Maroon" title="c24">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E81">A1058</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E82">A1059</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c27">q1</font>,<font color="Maroon" title="c28">q2</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E87">A1064</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E90">A1067</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E88">A1065</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>
;<br/>
<a NAME="E20:2_1_1_2_7_1_2_1_1_3_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A1069</font></span>: 
 ex <font color="Olive" title="b1">tm</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">tm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> &amp; <font color="Olive" title="b1">tm</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">tm</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">tm</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E91">A1068</font></span></a></span>;<br/>
<a NAME="E21:2_1_1_2_7_1_2_1_1_3_7"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E79">A1056</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E80">A1057</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c25">p1</font>,<font color="Maroon" title="c26">p2</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E85">A1062</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E84">A1061</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E86">A1063</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>
;<br/>
<a NAME="E23:2_1_1_2_7_1_2_1_1_3_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A1070</font></span>: 
 ex <font color="Olive" title="b1">tn</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">tn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">m</font>)</span>)</span> &amp; <font color="Olive" title="b1">tn</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c25">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">tn</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">tn</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E89">A1066</font></span></a></span>;<br/>
<a NAME="E24:2_1_1_2_7_1_2_1_1_3_7"/>
<font color="Maroon" title="c26">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A1045</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E78">A1055</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E80">A1057</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E81">A1058</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:2_1_1_2_7_1_2_1_1_3_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E93">A1070</font></span></a>, <a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_3_7"><span class="lab"><font color="Green" title="E92">A1069</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E25:2_1_1_2_7_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A1071</font></span>: 
<font color="Maroon" title="c20">g2</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#T1" onmouseover="rs('goboard2/T1')" onmouseout="rh()">GOBOARD2:1</a></span>;<br/><a NAME="E26:2_1_1_2_7_1_2_1_1_3"/><span class="lab"><font color="Green" title="E79">A1072</font></span>: 
 not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c23">ls</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span> } </span> ;<br/>
<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_3_8"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_8"/><span class="lab"><font color="Green" title="E80">A1073</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c24">X</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_8"/><span class="lab"><font color="Green" title="E81">A1074</font></span>: 
<font color="Maroon" title="c24">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c25">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_8"/><span class="lab"><font color="Green" title="E82">A1075</font></span>: 
<font color="Maroon" title="c24">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c25">m</font>)
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_8"/><span class="lab"><font color="Green" title="E83">A1076</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">m</font> &amp; <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_8"><span class="lab"><font color="Green" title="E81">A1074</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">m</font>, <font color="Maroon" title="c27">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E9:2_1_1_2_7_1_2_1_1_3_8"/><span class="lab"><font color="Green" title="E84">A1077</font></span>: 
<font color="Maroon" title="c25">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_8"><span class="lab"><font color="Green" title="E83">A1076</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E10:2_1_1_2_7_1_2_1_1_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A1078</font></span>: 
<font color="Maroon" title="c26">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E73">A1027</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c28">lq</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/>
<a NAME="E11:2_1_1_2_7_1_2_1_1_3_8"/><span class="lab"><font color="Green" title="E86">A1079</font></span>: 
( <font color="Maroon" title="c26">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c27">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c27">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_2_1_1_3_8"/><span class="lab"><font color="Green" title="E87">A1080</font></span>: 
<font color="Maroon" title="c25">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_8"><span class="lab"><font color="Green" title="E83">A1076</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_2_1_1_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A1081</font></span>: 
<font color="Maroon" title="c27">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E73">A1027</font></span></a></span>;<br/>
<a NAME="E14:2_1_1_2_7_1_2_1_1_3_8"/>
<font color="Maroon" title="c25">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E15:2_1_1_2_7_1_2_1_1_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A1082</font></span>: 
<font color="Maroon" title="c26">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A1045</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_8"><span class="lab"><font color="Green" title="E84">A1077</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3_8"><span class="lab"><font color="Green" title="E87">A1080</font></span></a></span>;<br/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c25">m</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c26">q1</font>,<font color="Maroon" title="c27">q2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_8"><span class="lab"><font color="Green" title="E83">A1076</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_8"><span class="lab"><font color="Green" title="E85">A1078</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3_8"><span class="lab"><font color="Green" title="E88">A1081</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_3_8"><span class="lab"><font color="Green" title="E89">A1082</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3_8"><span class="lab"><font color="Green" title="E86">A1079</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>
;<br/>
<a NAME="E17:2_1_1_2_7_1_2_1_1_3_8"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c26">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_8"><span class="lab"><font color="Green" title="E80">A1073</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_8"><span class="lab"><font color="Green" title="E82">A1075</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:2_1_1_2_7_1_2_1_1_3_8"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E73">A1027</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_8"><span class="lab"><font color="Green" title="E84">A1077</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E27:2_1_1_2_7_1_2_1_1_3"/>
<span class="p1">(<span class="default"><a href="goboard1.html#K2" title="GOBOARD1:func.2">Y_axis</a> <font color="Maroon" title="c11">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="txt" href="#E45:2_1_1_2"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="txt" href="#E46:2_1_1_2"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E43:2_1_1_2"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E65">A1014</font></span></a>, <a class="ref" href="goboard1.html#D2" onmouseover="rs('goboard1/D2')" onmouseout="rh()">GOBOARD1:def 2</a></span>;<br/><a NAME="E28:2_1_1_2_7_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A1083</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="txt" href="#E52:2_1_1_2"><span class="lab"><font color="Green" title="E55">A70</font></span></a>, <a class="txt" href="#E45:2_1_1_2"><span class="lab"><font color="Green" title="E48">A63</font></span></a>, <a class="txt" href="#E46:2_1_1_2"><span class="lab"><font color="Green" title="E49">A64</font></span></a>, <a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E43:2_1_1_2"><span class="lab"><font color="Green" title="E46">A61</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E72">A1026</font></span></a>, <a class="ref" href="seqm_3.html#D10" onmouseover="rs('seqm_3/D10')" onmouseout="rh()">SEQM_3:def 10</a></span>;<br/><div><span class="lab"><font color="Green" title="E81">A1084</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_9"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> holds <br/> for <font color="Olive" title="b2">l1</font>, <font color="Olive" title="b3">l2</font>, <font color="Olive" title="b4">l3</font>, <font color="Olive" title="b5">l4</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">l1</font>,<font color="Olive" title="b3">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">l3</font>,<font color="Olive" title="b5">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">l1</font>,<font color="Olive" title="b3">l2</font>) &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b4">l3</font>,<font color="Olive" title="b5">l4</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> implies  for <font color="Olive" title="b1">l1</font>, <font color="Olive" title="b2">l2</font>, <font color="Olive" title="b3">l3</font>, <font color="Olive" title="b4">l4</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font>) &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_3_9"/><span class="lab"><font color="Green" title="E82">A1085</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_3_9"/><span class="lab"><font color="Green" title="E83">A1086</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">l1</font>, <font color="Olive" title="b2">l2</font>, <font color="Olive" title="b3">l3</font>, <font color="Olive" title="b4">l4</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font>) &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c24">l1</font>, <font color="Maroon" title="c25">l2</font>, <font color="Maroon" title="c26">l3</font>, <font color="Maroon" title="c27">l4</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">l1</font>,<font color="Maroon" title="c25">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">l3</font>,<font color="Maroon" title="c27">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c24">l1</font>,<font color="Maroon" title="c25">l2</font>) &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c26">l3</font>,<font color="Maroon" title="c27">l4</font>) implies <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_9"/><span class="lab"><font color="Green" title="E84">A1087</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">l1</font>,<font color="Maroon" title="c25">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_9"/><span class="lab"><font color="Green" title="E85">A1088</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">l3</font>,<font color="Maroon" title="c27">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_9"/><span class="lab"><font color="Green" title="E86">A1089</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c24">l1</font>,<font color="Maroon" title="c25">l2</font>)
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_9"/><span class="lab"><font color="Green" title="E87">A1090</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c26">l3</font>,<font color="Maroon" title="c27">l4</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_9"/>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">i2</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E69">A1018</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E70">A1023</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_9"><span class="lab"><font color="Green" title="E83">A1086</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A1091</font></span>: 
( <font color="Maroon" title="c26">l3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c27">l4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_9"><span class="lab"><font color="Green" title="E85">A1088</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_9"><span class="lab"><font color="Green" title="E87">A1090</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_3_9"/>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E69">A1018</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E70">A1023</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_9"><span class="lab"><font color="Green" title="E82">A1085</font></span></a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_3_9"/><span class="kw">then </span>
( <font color="Maroon" title="c24">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c25">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_9"><span class="lab"><font color="Green" title="E84">A1087</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_9"><span class="lab"><font color="Green" title="E86">A1089</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> = 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c23">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_9"><span class="lab"><font color="Green" title="E88">A1091</font></span></a>, <a class="ref" href="absvalue.html#T2" onmouseover="rs('absvalue/T2')" onmouseout="rh()">ABSVALUE:2</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>

<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default">1</span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T52" onmouseover="rs('complex1/T52')" onmouseout="rh()">COMPLEX1:52</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_10"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">l1</font>, <font color="Olive" title="b2">l2</font>, <font color="Olive" title="b3">l3</font>, <font color="Olive" title="b4">l4</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b1">l1</font>,<font color="Olive" title="b2">l2</font>) &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b3">l3</font>,<font color="Olive" title="b4">l4</font>) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font> &amp; 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c23">l1</font>, <font color="Maroon" title="c24">l2</font>, <font color="Maroon" title="c25">l3</font>, <font color="Maroon" title="c26">l4</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">l1</font>,<font color="Maroon" title="c24">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">l3</font>,<font color="Maroon" title="c26">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c23">l1</font>,<font color="Maroon" title="c24">l2</font>) &amp; <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">l3</font>,<font color="Maroon" title="c26">l4</font>) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font> &amp; 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font> implies <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c25">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_3_10"/><span class="lab"><font color="Green" title="E82">A1092</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">l1</font>,<font color="Maroon" title="c24">l2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_3_10"/><span class="lab"><font color="Green" title="E83">A1093</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">l3</font>,<font color="Maroon" title="c26">l4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">and </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_10"/><span class="lab"><font color="Green" title="E84">A1094</font></span>: 
<font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c23">l1</font>,<font color="Maroon" title="c24">l2</font>)
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_10"/><span class="lab"><font color="Green" title="E85">A1095</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c25">l3</font>,<font color="Maroon" title="c26">l4</font>)
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_10"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_10"/><span class="lab"><font color="Green" title="E86">A1096</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c25">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_10"/>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c9">i2</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E69">A1018</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E70">A1023</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_10"><span class="lab"><font color="Green" title="E86">A1096</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A1097</font></span>: 
( <font color="Maroon" title="c25">l3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">l4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_10"><span class="lab"><font color="Green" title="E83">A1093</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_10"><span class="lab"><font color="Green" title="E85">A1095</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_3_10"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_3_10"/><span class="kw">then </span>
( <font color="Maroon" title="c23">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> &amp; <font color="Maroon" title="c24">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E20:2_1_1_2"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_10"><span class="lab"><font color="Green" title="E82">A1092</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_10"><span class="lab"><font color="Green" title="E84">A1094</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">l1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c25">l3</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">l2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">l4</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> = 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_10"><span class="lab"><font color="Green" title="E87">A1097</font></span></a>, <a class="ref" href="absvalue.html#T2" onmouseover="rs('absvalue/T2')" onmouseout="rh()">ABSVALUE:2</a></span>
<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>

<br/>.= 
<span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default">1</span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T52" onmouseover="rs('complex1/T52')" onmouseout="rh()">COMPLEX1:52</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" onmouseover="rs('absvalue/D1')" onmouseout="rh()">ABSVALUE:def 1</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E31:2_1_1_2_7_1_2_1_1_3"/><span class="kw">then </span>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/> for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font>) &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">i</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">j</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:2_1_1_2"><span class="lab"><font color="Green" title="E40">A48</font></span></a>, <a class="txt" href="#E29:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E81">A1084</font></span></a>, <a class="ref" href="goboard1.html#T24" onmouseover="rs('goboard1/T24')" onmouseout="rh()">GOBOARD1:24</a></span>;<br/><span class="kw">hence </span><a NAME="E32:2_1_1_2_7_1_2_1_1_3"/>
<font color="Maroon" title="c21">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E71">A1025</font></span></a>, <a class="ref" href="goboard1.html#D9" onmouseover="rs('goboard1/D9')" onmouseout="rh()">GOBOARD1:def 9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )</span><br/><a NAME="E33:2_1_1_2_7_1_2_1_1_3"/><span class="lab"><font color="Green" title="E82">A1098</font></span>: 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><span class="lab"><font color="Green" title="E83">A1099</font></span>:  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E12:2_1_1_2"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2"><span class="lab"><font color="Green" title="E61">A698</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E67">A1016</font></span></a>, <a class="txt" href="#E28:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E80">A1083</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E76">A1044</font></span></a>, <a class="txt" href="#E33:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E82">A1098</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>
;<br/><a NAME="E35:2_1_1_2_7_1_2_1_1_3"/><span class="lab"><font color="Green" title="E84">A1100</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font> or <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_3_12"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_12"/><span class="lab"><font color="Green" title="E85">A1101</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_12"/><span class="lab"><font color="Green" title="E86">A1102</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">pn</font> = <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>);<br/>
<a NAME="E5:2_1_1_2_7_1_2_1_1_3_12"/><span class="lab"><font color="Green" title="E87">A1103</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_12"><span class="lab"><font color="Green" title="E85">A1101</font></span></a></span>;<br/>
<a NAME="E6:2_1_1_2_7_1_2_1_1_3_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A1104</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E73">A1027</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_12"><span class="lab"><font color="Green" title="E85">A1101</font></span></a></span>;<br/>
<a NAME="E7:2_1_1_2_7_1_2_1_1_3_12"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E73">A1027</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_12"><span class="lab"><font color="Green" title="E85">A1101</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_12"><span class="lab"><font color="Green" title="E87">A1103</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:2_1_1_2_7_1_2_1_1_3_12"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E83">A1099</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_12"><span class="lab"><font color="Green" title="E86">A1102</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_12"><span class="lab"><font color="Green" title="E87">A1103</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_12"><span class="lab"><font color="Green" title="E88">A1104</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E36:2_1_1_2_7_1_2_1_1_3"/><span class="lab"><font color="Green" title="E85">A1105</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E37:2_1_1_2_7_1_2_1_1_3"/><span class="lab"><font color="Green" title="E86">A1106</font></span>: 
 not <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_3_13"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_13"/><span class="lab"><font color="Green" title="E87">A1107</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_13"/><span class="lab"><font color="Green" title="E88">A1108</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E5:2_1_1_2_7_1_2_1_1_3_13"/>
( <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">n</font></span>)</span>,<font color="Maroon" title="c9">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E36:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E85">A1105</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E69">A1018</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_13"><span class="lab"><font color="Green" title="E87">A1107</font></span></a></span>;<br/>
<a NAME="E6:2_1_1_2_7_1_2_1_1_3_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A1109</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_13"><span class="lab"><font color="Green" title="E88">A1108</font></span></a>, <a class="ref" href="goboard1.html#T5" onmouseover="rs('goboard1/T5')" onmouseout="rh()">GOBOARD1:5</a></span>;<br/>
<a NAME="E7:2_1_1_2_7_1_2_1_1_3_13"/>
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_13"><span class="lab"><font color="Green" title="E87">A1107</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:2_1_1_2_7_1_2_1_1_3_13"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_13"><span class="lab"><font color="Green" title="E89">A1109</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E38:2_1_1_2_7_1_2_1_1_3"/>
<span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_14"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c23">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_3_14"/><span class="lab"><font color="Green" title="E87">A1110</font></span>: 
 not <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:2_1_1_2_7_1_2_1_1_3_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A1111</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:2_1_1_2_7_1_2_1_1_3_14"/><span class="lab"><font color="Green" title="E89">A1112</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_14"><span class="lab"><font color="Green" title="E87">A1110</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_14_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_14_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_3_14_1_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_14_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_14_1_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_1_2_7_1_2_1_1_3_14_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E37:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E86">A1106</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_14"><span class="lab"><font color="Green" title="E89">A1112</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_14"><span class="lab"><font color="Green" title="E88">A1111</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_14_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_14_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_14_1_1_2"/><span class="kw">then </span>
(  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E38:2_1_1_2"><span class="lab"><font color="Green" title="E41">A49</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E84">A1100</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_14"><span class="lab"><font color="Green" title="E89">A1112</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_14"><span class="lab"><font color="Green" title="E88">A1111</font></span></a>, <a class="ref" href="goboard2.html#T4" onmouseover="rs('goboard2/T4')" onmouseout="rh()">GOBOARD2:4</a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_2_1_1_3_14_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E86">A1106</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_14"><span class="lab"><font color="Green" title="E88">A1111</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_2_1_1_3_14"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E39:2_1_1_2_7_1_2_1_1_3"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">g1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 ;<br/><span class="kw">hence </span><a NAME="E40:2_1_1_2_7_1_2_1_1_3"/>
<font color="Maroon" title="c21">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:2_1_1_2"><span class="lab"><font color="Green" title="E32">A40</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E75">A1043</font></span></a>, <a class="ref" href="finseq_3.html#T91" onmouseover="rs('finseq_3/T91')" onmouseout="rh()">FINSEQ_3:91</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )</span><br/><a NAME="E41:2_1_1_2_7_1_2_1_1_3"/><span class="lab"><font color="Green" title="E87">A1113</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E12:2_1_1_2"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2"><span class="lab"><font color="Green" title="E61">A698</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E42:2_1_1_2_7_1_2_1_1_3"/><span class="lab"><font color="Green" title="E88">A1114</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> holds <br/><span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_15"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> &amp; <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> implies <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_3_15"/><span class="lab"><font color="Green" title="E89">A1115</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_3_15"/><span class="lab"><font color="Green" title="E90">A1116</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E3:2_1_1_2_7_1_2_1_1_3_15"/><span class="lab"><font color="Green" title="E91">A1117</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E89">A1115</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E90">A1116</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/>
<a NAME="E4:2_1_1_2_7_1_2_1_1_3_15"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E5:2_1_1_2_7_1_2_1_1_3_15"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E83">A1099</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E84">A1100</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_15"/><span class="lab"><font color="Green" title="E92">A1118</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">u1</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_15"/><span class="lab"><font color="Green" title="E93">A1119</font></span>: 
<font color="Maroon" title="c24">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E9:2_1_1_2_7_1_2_1_1_3_15"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E10:2_1_1_2_7_1_2_1_1_3_15"/>
<font color="Maroon" title="c24">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ;<br/>
<a NAME="E11:2_1_1_2_7_1_2_1_1_3_15"/><span class="lab"><font color="Green" title="E94">A1120</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E89">A1115</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E90">A1116</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_2_1_1_3_15"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_2_1_1_3_15"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E83">A1099</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E84">A1100</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">u2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E15:2_1_1_2_7_1_2_1_1_3_15"/><span class="lab"><font color="Green" title="E95">A1121</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u2</font>
 <span class="kw">and </span><br/><a NAME="E16:2_1_1_2_7_1_2_1_1_3_15"/><span class="lab"><font color="Green" title="E96">A1122</font></span>: 
<font color="Maroon" title="c25">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E17:2_1_1_2_7_1_2_1_1_3_15"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E18:2_1_1_2_7_1_2_1_1_3_15"/>
<font color="Maroon" title="c25">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ;<br/>
<a NAME="E19:2_1_1_2_7_1_2_1_1_3_15"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E20:2_1_1_2_7_1_2_1_1_3_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A1123</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c24">u1</font>,<font color="Maroon" title="c25">u2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E90">A1116</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E92">A1118</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E95">A1121</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E21:2_1_1_2_7_1_2_1_1_3_15"/>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E22:2_1_1_2_7_1_2_1_1_3_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A1124</font></span>: 
<font color="Maroon" title="c24">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A1045</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E91">A1117</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E94">A1120</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E92">A1118</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E95">A1121</font></span></a></span>;<br/>
<a NAME="E23:2_1_1_2_7_1_2_1_1_3_15"/><span class="lab"><font color="Green" title="E99">A1125</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E89">A1115</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E90">A1116</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/>
<a NAME="E24:2_1_1_2_7_1_2_1_1_3_15"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E25:2_1_1_2_7_1_2_1_1_3_15"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E83">A1099</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E84">A1100</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E27:2_1_1_2_7_1_2_1_1_3_15"/><span class="lab"><font color="Green" title="E100">A1126</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u</font>
 <span class="kw">and </span><br/><a NAME="E28:2_1_1_2_7_1_2_1_1_3_15"/><span class="lab"><font color="Green" title="E101">A1127</font></span>: 
<font color="Maroon" title="c26">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E29:2_1_1_2_7_1_2_1_1_3_15"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E30:2_1_1_2_7_1_2_1_1_3_15"/>
<font color="Maroon" title="c26">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ;<br/>
<a NAME="E31:2_1_1_2_7_1_2_1_1_3_15"/>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E32:2_1_1_2_7_1_2_1_1_3_15"/><span class="kw">then </span>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E90">A1116</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E33:2_1_1_2_7_1_2_1_1_3_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A1128</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c26">u</font>,<font color="Maroon" title="c24">u1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E89">A1115</font></span></a>, <a class="txt" href="#E27:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E100">A1126</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E92">A1118</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c27">lg</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c26">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/>
<a NAME="E34:2_1_1_2_7_1_2_1_1_3_15"/>
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E35:2_1_1_2_7_1_2_1_1_3_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A1129</font></span>: 
<font color="Maroon" title="c26">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A1045</font></span></a>, <a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E99">A1125</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E91">A1117</font></span></a>, <a class="txt" href="#E27:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E100">A1126</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E92">A1118</font></span></a></span>;<br/>
<a NAME="E36:2_1_1_2_7_1_2_1_1_3_15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A1130</font></span>: 
<font color="Maroon" title="c24">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c26">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E93">A1119</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E98">A1124</font></span></a></span>;<br/>
<a NAME="E37:2_1_1_2_7_1_2_1_1_3_15"/>
( <font color="Maroon" title="c26">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">u</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c25">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<a NAME="E38:2_1_1_2_7_1_2_1_1_3_15"/><span class="kw">then </span>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c26">u</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E100">A1126</font></span></a>, <a class="txt" href="#E28:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E101">A1127</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E95">A1121</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E96">A1122</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E98">A1124</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E103">A1129</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E39:2_1_1_2_7_1_2_1_1_3_15"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E100">A1126</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E92">A1118</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E95">A1121</font></span></a>, <a class="txt" href="#E33:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E102">A1128</font></span></a>, <a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E97">A1123</font></span></a>, <a class="txt" href="#E36:2_1_1_2_7_1_2_1_1_3_15"><span class="lab"><font color="Green" title="E104">A1130</font></span></a>, <a class="ref" href="topreal1.html#T8" onmouseover="rs('topreal1/T8')" onmouseout="rh()">TOPREAL1:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E43:2_1_1_2_7_1_2_1_1_3"/>
<font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_16"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="topreal1.html#D6" onmouseover="rs('topreal1/D6')" onmouseout="rh()">TOPREAL1:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> or  not <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> or <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_3_16"/><span class="lab"><font color="Green" title="E89">A1131</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_3_16"/><span class="lab"><font color="Green" title="E90">A1132</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E3:2_1_1_2_7_1_2_1_1_3_16"/><span class="lab"><font color="Green" title="E91">A1133</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E90">A1132</font></span></a></span>;<br/>
<a NAME="E4:2_1_1_2_7_1_2_1_1_3_16"/>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E5:2_1_1_2_7_1_2_1_1_3_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A1134</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E90">A1132</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E6:2_1_1_2_7_1_2_1_1_3_16"/><span class="lab"><font color="Green" title="E93">A1135</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<a NAME="E7:2_1_1_2_7_1_2_1_1_3_16"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2
 
;<br/>
<a NAME="E8:2_1_1_2_7_1_2_1_1_3_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A1136</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E90">A1132</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E93">A1135</font></span></a>, <a class="ref" href="xreal_1.html#T20" onmouseover="rs('xreal_1/T20')" onmouseout="rh()">XREAL_1:20</a></span>;<br/>
<a NAME="E9:2_1_1_2_7_1_2_1_1_3_16"/><span class="lab"><font color="Green" title="E95">A1137</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E10:2_1_1_2_7_1_2_1_1_3_16"/><span class="lab"><font color="Green" title="E96">A1138</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E11:2_1_1_2_7_1_2_1_1_3_16"/><span class="lab"><font color="Green" title="E97">A1139</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_16_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_3_16_1"/>
( <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_16_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_16_1_1"/><span class="lab"><font color="Green" title="E98">A1140</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_16_1_1"/><span class="lab"><font color="Green" title="E99">A1141</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_16_1_1"/><span class="lab"><font color="Green" title="E100">A1142</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E89">A1131</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E98">A1140</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_16_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A1143</font></span>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_16_1_1"/>
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E89">A1131</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E98">A1140</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_16_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A1144</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c23">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E100">A1142</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_16_1_1"/>
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E89">A1131</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E98">A1140</font></span></a>, <a class="ref" href="seq_4.html#T135" onmouseover="rs('seq_4/T135')" onmouseout="rh()">SEQ_4:135</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_16_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E100">A1142</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E99">A1141</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_2_1_1_3_16_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:2_1_1_2"><span class="lab"><font color="Green" title="E33">A41</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E89">A1131</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E98">A1140</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E102">A1144</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_16_1_1"><span class="lab"><font color="Green" title="E101">A1143</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_16_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_16_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_16_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_16_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A1145</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_16_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_16_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/><span class="lab"><font color="Green" title="E99">A1146</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E91">A1133</font></span></a>, <a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E93">A1135</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A1147</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E83">A1099</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E84">A1100</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/><span class="lab"><font color="Green" title="E101">A1148</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">u1</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/>
<font color="Maroon" title="c24">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E9:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E10:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/>
<font color="Maroon" title="c24">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/>
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T68" onmouseover="rs('rltopsp1/T68')" onmouseout="rh()">RLTOPSP1:68</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A1149</font></span>: 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<font color="Maroon" title="c24">u1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E87">A1113</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E84">A1100</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A1147</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A1148</font></span></a>, <a class="ref" href="topreal1.html#T6" onmouseover="rs('topreal1/T6')" onmouseout="rh()">TOPREAL1:6</a></span>;<br/><a NAME="E13:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E14:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A1150</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E99">A1146</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A1151</font></span>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E99">A1146</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><a NAME="E1:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E89">A1131</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E99">A1146</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E2:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A1152</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">assume </span><a NAME="E3:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E4:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E105">A1152</font></span></a>, <a class="ref" href="topreal1.html#T23" onmouseover="rs('topreal1/T23')" onmouseout="rh()">TOPREAL1:23</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E17:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E18:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A1153</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="goboard2.html#T4" onmouseover="rs('goboard2/T4')" onmouseout="rh()">GOBOARD2:4</a></span>;<br/><a NAME="E19:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/><span class="lab"><font color="Green" title="E106">A1154</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c23">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="ref" href="topreal3.html#T19" onmouseover="rs('topreal3/T19')" onmouseout="rh()">TOPREAL3:19</a></span>;<br/><a NAME="E20:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/>
<font color="Maroon" title="c23">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E89">A1131</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E96">A1138</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E99">A1146</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E21:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A1155</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c23">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E103">A1150</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E22:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/>
( <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>) &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E89">A1131</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E90">A1132</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E95">A1137</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E92">A1134</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E97">A1139</font></span></a>, <a class="ref" href="topreal1.html#T21" onmouseover="rs('topreal1/T21')" onmouseout="rh()">TOPREAL1:21</a></span>;<br/><a NAME="E23:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E24:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A1156</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/><a NAME="E25:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/>
( <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E90">A1132</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E97">A1139</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E93">A1135</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E99">A1146</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E26:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><a NAME="E27:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<font color="Maroon" title="c24">u1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E90">A1132</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E95">A1137</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E97">A1139</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E104">A1151</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A1148</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E28:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E106">A1154</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A1153</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E107">A1155</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E104">A1151</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A1149</font></span></a>, <a class="ref" href="xboole_1.html#T27" onmouseover="rs('xboole_1/T27')" onmouseout="rh()">XBOOLE_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E29:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_1"><span class="lab"><font color="Green" title="E108">A1156</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_16_1_2"><span class="lab"><font color="Green" title="E98">A1145</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A1157</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">n1</font> = <font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E100">A1158</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A1159</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E90">A1132</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E93">A1135</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A1160</font></span>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A1158</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A1159</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A1161</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E83">A1099</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E84">A1100</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">u1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E10:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E104">A1162</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u1</font>
 <span class="kw">and </span><br/><a NAME="E11:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E105">A1163</font></span>: 
( <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E12:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c25">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ;<br/><a NAME="E13:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E14:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c23">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A1158</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/><a NAME="E16:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A1164</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<font color="Maroon" title="c25">u1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E89">A1131</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E92">A1134</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A1160</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E104">A1162</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E17:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E107">A1165</font></span>: 
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A1159</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E18:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/><a NAME="E19:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E83">A1099</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E84">A1100</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">u2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E21:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E108">A1166</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">u2</font>
 <span class="kw">and </span><br/><a NAME="E22:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E109">A1167</font></span>: 
( <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E23:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c26">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ;<br/><span class="kw">set </span><font color="Maroon" title="c27">lg</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/><a NAME="E24:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c25">u1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A1045</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E103">A1161</font></span></a>, <a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E107">A1165</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E104">A1162</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E108">A1166</font></span></a></span>;<br/><a NAME="E25:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c26">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c26">u2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c25">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">w</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">w</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">u2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A1163</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><a NAME="E26:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A1168</font></span>: 
<font color="Maroon" title="c25">u1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<font color="Maroon" title="c26">u2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E76">A1044</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E109">A1167</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>;<br/><a NAME="E27:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E100">A1158</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E101">A1159</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><a NAME="E28:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c25">u1</font>,<font color="Maroon" title="c26">u2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E90">A1132</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E95">A1137</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E97">A1139</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A1160</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E104">A1162</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E108">A1166</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E29:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E102">A1160</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E104">A1162</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E106">A1164</font></span></a>, <a class="txt" href="#E26:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E110">A1168</font></span></a>, <a class="ref" href="topreal1.html#T8" onmouseover="rs('topreal1/T8')" onmouseout="rh()">TOPREAL1:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A1169</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2"><span class="lab"><font color="Green" title="E99">A1157</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A1170</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">n</font>
 ;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A1171</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c24">n1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E92">A1134</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E100">A1169</font></span></a>, <a class="ref" href="goboard2.html#T5" onmouseover="rs('goboard2/T5')" onmouseout="rh()">GOBOARD2:5</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E103">A1172</font></span>: 
<font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E92">A1134</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E93">A1135</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A1173</font></span>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E103">A1172</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E96">A1138</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E100">A1169</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E91">A1133</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E103">A1172</font></span></a>, <a class="ref" href="goboard2.html#T5" onmouseover="rs('goboard2/T5')" onmouseout="rh()">GOBOARD2:5</a></span>;<br/><span class="kw">hence </span><a NAME="E12:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E88">A1114</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_16"><span class="lab"><font color="Green" title="E94">A1136</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E102">A1171</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E104">A1173</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E101">A1170</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_2_1_1_3_16_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_2_1_1_3_16_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c23">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E44:2_1_1_2_7_1_2_1_1_3"/><span class="lab"><font color="Green" title="E89">A1174</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_17"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font> or <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c24">ls</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span> } </span> ;<br/>
<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_3_17"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c25">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_17"/><span class="lab"><font color="Green" title="E90">A1175</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">X</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_17"/><span class="lab"><font color="Green" title="E91">A1176</font></span>: 
<font color="Maroon" title="c25">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">m</font>)</span>)</span> where <font color="Olive" title="b1">m</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c26">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_17"/><span class="lab"><font color="Green" title="E92">A1177</font></span>: 
<font color="Maroon" title="c25">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c26">m</font>)
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_17"/><span class="lab"><font color="Green" title="E93">A1178</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">m</font> &amp; <font color="Maroon" title="c26">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_17"><span class="lab"><font color="Green" title="E91">A1176</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">m</font>, <font color="Maroon" title="c28">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E9:2_1_1_2_7_1_2_1_1_3_17"/><span class="lab"><font color="Green" title="E94">A1179</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c26">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c27">q1</font>,<font color="Maroon" title="c28">q2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_17"><span class="lab"><font color="Green" title="E93">A1178</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E10:2_1_1_2_7_1_2_1_1_3_17"/>
<font color="Maroon" title="c26">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_17"><span class="lab"><font color="Green" title="E93">A1178</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E11:2_1_1_2_7_1_2_1_1_3_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A1180</font></span>: 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_2_1_1_3_17"/>
<font color="Maroon" title="c26">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_17"><span class="lab"><font color="Green" title="E93">A1178</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_2_1_1_3_17"/><span class="kw">then </span>
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>;<br/>
<a NAME="E14:2_1_1_2_7_1_2_1_1_3_17"/><span class="kw">then </span>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c27">q1</font>,<font color="Maroon" title="c28">q2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E87">A1113</font></span></a>, <a class="txt" href="#E35:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E84">A1100</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3_17"><span class="lab"><font color="Green" title="E95">A1180</font></span></a>, <a class="ref" href="topreal1.html#T6" onmouseover="rs('topreal1/T6')" onmouseout="rh()">TOPREAL1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:2_1_1_2_7_1_2_1_1_3_17"/>
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E87">A1113</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_17"><span class="lab"><font color="Green" title="E90">A1175</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_17"><span class="lab"><font color="Green" title="E92">A1177</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_17"><span class="lab"><font color="Green" title="E94">A1179</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E45:2_1_1_2_7_1_2_1_1_3"/><span class="lab"><font color="Green" title="E90">A1181</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_18"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_18_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_3_18_1"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_18_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_18_1_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:2_1_1_2_7_1_2_1_1_3_18_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2"><span class="lab"><font color="Green" title="E43">A52</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_18_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_18_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_18_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_18_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="ref" href="goboard2.html#T4" onmouseover="rs('goboard2/T4')" onmouseout="rh()">GOBOARD2:4</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_18_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A1182</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E89">A1174</font></span></a>, <a class="ref" href="xboole_1.html#T26" onmouseover="rs('xboole_1/T26')" onmouseout="rh()">XBOOLE_1:26</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_18_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c23">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span>;<br/><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_3_18_1_2_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_3_18_1_2_1"/><span class="kw">then </span>
(  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_18_1_2"><span class="lab"><font color="Green" title="E91">A1182</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_2_1_1_3_18_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E79">A1072</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_2_1_1_3_18_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E46:2_1_1_2_7_1_2_1_1_3"/>
 for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> holds <br/> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Olive" title="b2">m</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E91">A1183</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E2:2_1_1_2_7_1_2_1_1_3_19"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E92">A1184</font></span>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> = 
<font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">qq</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c24">l1</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">l2</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span> } </span> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c26">n</font>, <font color="Maroon" title="c27">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c26">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c27">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> &amp; <font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font> implies  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E93">A1185</font></span>: 
<font color="Maroon" title="c27">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E94">A1186</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E95">A1187</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E96">A1188</font></span>: 
<font color="Maroon" title="c27">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">and </span><br/><a NAME="E9:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E97">A1189</font></span>: 
<font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/>

<a NAME="E10:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E98">A1190</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E94">A1186</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E11:2_1_1_2_7_1_2_1_1_3_19"/>
<font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E62">A1011</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E12:2_1_1_2_7_1_2_1_1_3_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A1191</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/>
<a NAME="E13:2_1_1_2_7_1_2_1_1_3_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A1192</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E14:2_1_1_2_7_1_2_1_1_3_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A1193</font></span>: 
( <font color="Maroon" title="c23">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c23">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E73">A1027</font></span></a></span>;<br/>
<a NAME="E15:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E102">A1194</font></span>: 
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">qq</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E99">A1191</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/>
<a NAME="E16:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E103">A1195</font></span>: 
<font color="Maroon" title="c23">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E73">A1027</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E100">A1192</font></span></a></span>;<br/>
<a NAME="E17:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E104">A1196</font></span>: 
<font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E97">A1189</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E18:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E105">A1197</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E97">A1189</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E19:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E106">A1198</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E95">A1187</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E20:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E107">A1199</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E95">A1187</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E21:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E108">A1200</font></span>: 
<font color="Maroon" title="c23">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">qq</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/>
<a NAME="E22:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E109">A1201</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E96">A1188</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c28">ql</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/>
<a NAME="E23:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E110">A1202</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E24:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E111">A1203</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<a NAME="E25:2_1_1_2_7_1_2_1_1_3_19"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E99">A1191</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E112">A1204</font></span>:  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>,<font color="Maroon" title="c23">qq</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E91">A1183</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E92">A1184</font></span></a>, <a class="txt" href="#E15:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E102">A1194</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">z</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">z</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E76">A1044</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E101">A1193</font></span></a>, <a class="txt" href="#E21:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E108">A1200</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>
;<br/>
<a NAME="E27:2_1_1_2_7_1_2_1_1_3_19"/><span class="lab"><font color="Green" title="E113">A1205</font></span>: 
<font color="Maroon" title="c27">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/>
<a NAME="E28:2_1_1_2_7_1_2_1_1_3_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A1206</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E93">A1185</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_3_19_3_1"/>
( <font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_19_3_1_1"/><span class="lab"><font color="Green" title="E115">A1207</font></span>: 
<font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_19_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E113">A1205</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_19_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A1208</font></span>: 
<font color="Maroon" title="c27">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E109">A1201</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_19_3_1_1"/>
<font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E105">A1197</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_19_3_1_1"><span class="lab"><font color="Green" title="E115">A1207</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_19_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A1209</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c27">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_19_3_1_1"><span class="lab"><font color="Green" title="E116">A1208</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_19_3_1_1"/><span class="lab"><font color="Green" title="E118">A1210</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E114">A1206</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_19_3_1_1"><span class="lab"><font color="Green" title="E115">A1207</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_19_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c26">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E110">A1202</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_19_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A1211</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E98">A1190</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_3_19_3_1_1"/>
<font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E106">A1198</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_19_3_1_1"><span class="lab"><font color="Green" title="E118">A1210</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_3_19_3_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c26">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_19_3_1_1"><span class="lab"><font color="Green" title="E119">A1211</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><span class="kw">hence </span><a NAME="E11:2_1_1_2_7_1_2_1_1_3_19_3_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:2_1_1_2"><span class="lab"><font color="Green" title="E34">A42</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E93">A1185</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_19_3_1_1"><span class="lab"><font color="Green" title="E117">A1209</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_19_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A1212</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">m1</font> = <font color="Maroon" title="c27">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1"/>
( <font color="Maroon" title="c27">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or <font color="Maroon" title="c27">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E116">A1213</font></span>: 
<font color="Maroon" title="c27">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E117">A1214</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c30">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>) or <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c31">px</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E118">A1215</font></span>: 
( <font color="Maroon" title="c31">px</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">x</font> &amp; <font color="Maroon" title="c31">px</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c31">px</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E119">A1216</font></span>: 
<font color="Maroon" title="c31">px</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E112">A1204</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E116">A1213</font></span></a></span>;<br/>
<a NAME="E5:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c31">px</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E103">A1195</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E119">A1216</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_1"/>
<font color="Maroon" title="c30">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E83">A1099</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E118">A1215</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"/>
<font color="Maroon" title="c26">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E93">A1185</font></span></a>, <a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E110">A1202</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E116">A1213</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A1217</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E98">A1190</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span></a><div class="add"><a NAME="E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E93">A1185</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E106">A1198</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E116">A1213</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E2:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A1218</font></span>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">assume </span><a NAME="E3:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:2_1_1_2"><span class="lab"><font color="Green" title="E43">A52</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E119">A1218</font></span></a>, <a class="ref" href="topreal1.html#T23" onmouseover="rs('topreal1/T23')" onmouseout="rh()">TOPREAL1:23</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A1219</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="goboard2.html#T4" onmouseover="rs('goboard2/T4')" onmouseout="rh()">GOBOARD2:4</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"/><span class="lab"><font color="Green" title="E120">A1220</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E93">A1185</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E106">A1198</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E116">A1213</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A1221</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c26">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E118">A1217</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E93">A1185</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E98">A1190</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E116">A1213</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A1222</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E119">A1219</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E117">A1214</font></span></a>, <a class="ref" href="xboole_1.html#T27" onmouseover="rs('xboole_1/T27')" onmouseout="rh()">XBOOLE_1:27</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c30">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span>;<br/><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_3"/><span class="lab"><font color="Green" title="E123">A1223</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A1224</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_3"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E122">A1222</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E123">A1223</font></span></a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A1225</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_3"/>
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>;<br/><span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:2_1_1_2"><span class="lab"><font color="Green" title="E32">A40</font></span></a>, <a class="txt" href="#E30:2_1_1_2"><span class="lab"><font color="Green" title="E33">A41</font></span></a>, <a class="txt" href="#E31:2_1_1_2"><span class="lab"><font color="Green" title="E34">A42</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E93">A1185</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E116">A1213</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E118">A1217</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E120">A1220</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"><span class="lab"><font color="Green" title="E121">A1221</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E124">A1224</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E125">A1225</font></span></a>, <a class="ref" href="goboard2.html#T2" onmouseover="rs('goboard2/T2')" onmouseout="rh()">GOBOARD2:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E14:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"/>
<font color="Maroon" title="c27">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A1226</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c27">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_19_3_1_2"><span class="lab"><font color="Green" title="E115">A1212</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E117">A1227</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"/>
<font color="Maroon" title="c27">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A1228</font></span>: 
<font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E104">A1196</font></span></a>, <a class="txt" href="#E24:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E111">A1203</font></span></a>, <a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"/>
<font color="Maroon" title="c27">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E119">A1229</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c29">m1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E104">A1196</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E116">A1226</font></span></a>, <a class="ref" href="goboard2.html#T5" onmouseover="rs('goboard2/T5')" onmouseout="rh()">GOBOARD2:5</a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E117">A1227</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A1228</font></span></a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E120">A1230</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E121">A1231</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c26">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E110">A1202</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E122">A1232</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E98">A1190</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E106">A1198</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E121">A1231</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c26">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E122">A1232</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E98">A1190</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E121">A1231</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E90">A1181</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E120">A1230</font></span></a>, <a class="ref" href="xboole_1.html#T3" onmouseover="rs('xboole_1/T3')" onmouseout="rh()">XBOOLE_1:3</a>, <a class="ref" href="xboole_1.html#T27" onmouseover="rs('xboole_1/T27')" onmouseout="rh()">XBOOLE_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E121">A1233</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">n1</font> = <font color="Maroon" title="c26">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E122">A1234</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E123">A1235</font></span>: 
<font color="Maroon" title="c26">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">n</font> or <font color="Maroon" title="c26">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E124">A1236</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c31">q1</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">m1</font>, <font color="Maroon" title="c32">q2</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c33">x</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c34">q1l</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c31">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">v</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/><a NAME="E2:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E125">A1237</font></span>: 
( <font color="Maroon" title="c31">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c31">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp; <font color="Maroon" title="c32">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c32">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><span class="kw">assume </span><a NAME="E3:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E126">A1238</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A1239</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E126">A1238</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A1240</font></span>: 
 ex <font color="Olive" title="b1">qx</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">qx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> &amp; <font color="Olive" title="b1">qx</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qx</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">qx</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E112">A1204</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E124">A1236</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E129">A1241</font></span>: 
<font color="Maroon" title="c29">m1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E117">A1227</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A1228</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A1242</font></span>: 
<font color="Maroon" title="c31">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E73">A1027</font></span></a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E131">A1243</font></span>: 
<font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E117">A1227</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A1228</font></span></a>, <a class="ref" href="seq_4.html#T134" onmouseover="rs('seq_4/T134')" onmouseout="rh()">SEQ_4:134</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A1244</font></span>: 
<font color="Maroon" title="c32">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E73">A1027</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A1245</font></span>: 
<font color="Maroon" title="c31">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c32">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A1045</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E129">A1241</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E131">A1243</font></span></a></span>;<br/> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c29">m1</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c31">q1</font>,<font color="Maroon" title="c32">q2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E117">A1227</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E118">A1228</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">v</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c31">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">v</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E130">A1242</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E132">A1244</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E133">A1245</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E125">A1237</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>
;<br/><a NAME="E14:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E134">A1246</font></span>: 
 ex <font color="Olive" title="b1">qy</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">qy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span>)</span> &amp; <font color="Olive" title="b1">qy</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c31">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">qy</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">qy</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c32">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E119">A1229</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E127">A1239</font></span></a></span>;<br/><a NAME="E15:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
( <font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E93">A1185</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E122">A1234</font></span></a>, <a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E16:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E17:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">qq</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c31">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E77">A1045</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E100">A1192</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E129">A1241</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E18:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E128">A1240</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E134">A1246</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c26">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E121">A1233</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E124">A1247</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c20">g2</font>,<font color="Maroon" title="c30">n1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E107">A1199</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E123">A1235</font></span></a>, <a class="ref" href="goboard2.html#T5" onmouseover="rs('goboard2/T5')" onmouseout="rh()">GOBOARD2:5</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c29">m1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c30">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_19"><span class="lab"><font color="Green" title="E93">A1185</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E122">A1234</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E78">A1071</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"><span class="lab"><font color="Green" title="E119">A1229</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E124">A1247</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E12:2_1_1_2_7_1_2_1_1_3_19_3_1_2_1_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_2_1_1_3_19_3_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E30:2_1_1_2_7_1_2_1_1_3_19"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c26">n</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">m</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E47:2_1_1_2_7_1_2_1_1_3"/>
<font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#T1" onmouseover="rs('goboard2/T1')" onmouseout="rh()">GOBOARD2:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_20"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )</span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c23">p</font> = <font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c24">q</font> = <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1;<br/><a NAME="E1:2_1_1_2_7_1_2_1_1_3_20"/>
<font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E62">A1011</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E2:2_1_1_2_7_1_2_1_1_3_20"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_20"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_20"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E73">A1027</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_2_1_1_3_20"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E49:2_1_1_2_7_1_2_1_1_3"/>
<font color="Maroon" title="c21">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:2_1_1_2"><span class="lab"><font color="Green" title="E35">A43</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E74">A1036</font></span></a>, <a class="ref" href="goboard2.html#T8" onmouseover="rs('goboard2/T8')" onmouseout="rh()">GOBOARD2:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )</span><br/><span class="kw">thus </span><a NAME="E50:2_1_1_2_7_1_2_1_1_3"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c23">lg</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c24">lf</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span> } </span> ;<br/>
<a NAME="E1:2_1_1_2_7_1_2_1_1_3_21"/><span class="lab"><font color="Green" title="E91">A1248</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E92">A1249</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> holds <br/> for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font> holds <br/>( <font color="Olive" title="b2">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b2">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b2">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c25">j</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font> &amp; <font color="Maroon" title="c25">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> implies  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font> holds <br/>( <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_1"/><span class="lab"><font color="Green" title="E93">A1250</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_3_21_1"/><span class="lab"><font color="Green" title="E94">A1251</font></span>: 
<font color="Maroon" title="c25">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font> holds <br/>( <font color="Olive" title="b1">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c26">w</font> = <font color="Maroon" title="c25">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_1"><span class="lab"><font color="Green" title="E93">A1250</font></span></a>, <a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c27">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font> implies ( <font color="Maroon" title="c27">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> ) )</span><br/><span class="kw">assume </span><a NAME="E4:2_1_1_2_7_1_2_1_1_3_21_1"/><span class="lab"><font color="Green" title="E95">A1252</font></span>: 
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> )</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_1_1_1"/>
( <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or <font color="Maroon" title="c25">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_1_1_1_1"/><span class="lab"><font color="Green" title="E96">A1253</font></span>: 
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> )</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_21_1_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_21_1_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A1254</font></span>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:2_1_1_2_7_1_2_1_1_3_21_1_1_1_1"/>
<font color="Maroon" title="c27">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_21_1"><span class="lab"><font color="Green" title="E95">A1252</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_1_1_1_1"><span class="lab"><font color="Green" title="E96">A1253</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> )</span><br/><span class="kw">thus </span><a NAME="E6:2_1_1_2_7_1_2_1_1_3_21_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E14:2_1_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="txt" href="#E51:2_1_1_2"><span class="lab"><font color="Green" title="E54">A69</font></span></a>, <a class="txt" href="#E47:2_1_1_2"><span class="lab"><font color="Green" title="E50">A65</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E62">A1011</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E64">A1013</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E66">A1015</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_21_1"><span class="lab"><font color="Green" title="E95">A1252</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_1_1_1_1"><span class="lab"><font color="Green" title="E96">A1253</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_21_1_1_1_1"><span class="lab"><font color="Green" title="E97">A1254</font></span></a>, <a class="ref" href="seqm_3.html#D1" onmouseover="rs('seqm_3/D1')" onmouseout="rh()">SEQM_3:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c27">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font></span><br/><span class="kw">thus </span><a NAME="E7:2_1_1_2_7_1_2_1_1_3_21_1_1_1_1"/>
<font color="Maroon" title="c27">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:2_1_1_2"><span class="lab"><font color="Green" title="E51">A66</font></span></a>, <a class="txt" href="#E42:2_1_1_2"><span class="lab"><font color="Green" title="E45">A60</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E63">A1012</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_21_1"><span class="lab"><font color="Green" title="E95">A1252</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_1_1_1_1"><span class="lab"><font color="Green" title="E96">A1253</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_21_1_1_1_1"><span class="lab"><font color="Green" title="E97">A1254</font></span></a>, <a class="ref" href="partfun2.html#T2" onmouseover="rs('partfun2/T2')" onmouseout="rh()">PARTFUN2:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_1_1_1_2"/>
<font color="Maroon" title="c25">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> )</span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_21_1_1_1_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_1"><span class="lab"><font color="Green" title="E93">A1250</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_21_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_21_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A1255</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_21_1_1_1_2"/><span class="lab"><font color="Green" title="E97">A1256</font></span>: 
<font color="Maroon" title="c26">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21"><span class="lab"><font color="Green" title="E91">A1248</font></span></a>, <a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_21_1"><span class="lab"><font color="Green" title="E94">A1251</font></span></a>, <a class="ref" href="xreal_1.html#T20" onmouseover="rs('xreal_1/T20')" onmouseout="rh()">XREAL_1:20</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_21_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A1257</font></span>: 
<font color="Maroon" title="c26">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_21_1_1_1_2"><span class="lab"><font color="Green" title="E96">A1255</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_21_1_1_1_2"/>
<font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">w</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span>
 ;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_21_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c26">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_21_1_1_1_2"><span class="lab"><font color="Green" title="E96">A1255</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_21_1_1_1_2"><span class="lab"><font color="Green" title="E97">A1256</font></span></a>, <a class="ref" href="seq_4.html#T136" onmouseover="rs('seq_4/T136')" onmouseout="rh()">SEQ_4:136</a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_2_1_1_3_21_1_1_1_2"/>
( <font color="Maroon" title="c27">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E73">A1027</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_21_1"><span class="lab"><font color="Green" title="E95">A1252</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_21_1_1_1_2"><span class="lab"><font color="Green" title="E98">A1257</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_2_1_1_3_21_1"/>
( <font color="Maroon" title="c27">p</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c27">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c11">c1</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E3:2_1_1_2_7_1_2_1_1_3_21"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> or <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_21_2"/><span class="lab"><font color="Green" title="E93">A1258</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">X</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_21_2"/><span class="lab"><font color="Green" title="E94">A1259</font></span>: 
<font color="Maroon" title="c26">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_21_2"/><span class="lab"><font color="Green" title="E95">A1260</font></span>: 
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font>)
 <span class="kw">and </span><br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_21_2"/><span class="lab"><font color="Green" title="E96">A1261</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font>
 <span class="kw">and </span><br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_21_2"/><span class="lab"><font color="Green" title="E97">A1262</font></span>: 
<font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_21_2"><span class="lab"><font color="Green" title="E94">A1259</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_2_1_1"/>
( <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> or <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_2_1_1_1"/><span class="lab"><font color="Green" title="E98">A1263</font></span>: 
<font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_21_2_1_1_1"/>
<font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_21_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_2_1_1_1"><span class="lab"><font color="Green" title="E98">A1263</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_21_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A1264</font></span>: 
<font color="Maroon" title="c27">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_21_2"><span class="lab"><font color="Green" title="E96">A1261</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_21_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" onmouseover="rs('nat_1/T11')" onmouseout="rh()">NAT_1:11</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_21_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_2_1_1_1"><span class="lab"><font color="Green" title="E98">A1263</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_21_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c26">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c27">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_21_2"><span class="lab"><font color="Green" title="E95">A1260</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_21_2_1_1_1"><span class="lab"><font color="Green" title="E99">A1264</font></span></a>, <a class="ref" href="topreal3.html#T18" onmouseover="rs('topreal3/T18')" onmouseout="rh()">TOPREAL3:18</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_21_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c26">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c16">g1</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_21_2"><span class="lab"><font color="Green" title="E96">A1261</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_2_1_1_1"><span class="lab"><font color="Green" title="E98">A1263</font></span></a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_3_21_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A1265</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_21_2"><span class="lab"><font color="Green" title="E93">A1258</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_3_21_2_1_1_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T20" onmouseover="rs('topreal3/T20')" onmouseout="rh()">TOPREAL3:20</a></span>;<br/><span class="kw">hence </span><a NAME="E11:2_1_1_2_7_1_2_1_1_3_21_2_1_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_21_2_1_1_1"><span class="lab"><font color="Green" title="E100">A1265</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"/><span class="lab"><font color="Green" title="E98">A1266</font></span>: 
<font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c28">q1</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">i</font>, <font color="Maroon" title="c29">q2</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"/><span class="lab"><font color="Green" title="E99">A1267</font></span>: 
<font color="Maroon" title="c27">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_21_2"><span class="lab"><font color="Green" title="E97">A1262</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"/><span class="lab"><font color="Green" title="E100">A1268</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E98">A1266</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A1269</font></span>: 
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_21"><span class="lab"><font color="Green" title="E92">A1249</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E99">A1267</font></span></a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"/><span class="lab"><font color="Green" title="E102">A1270</font></span>: 
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_21"><span class="lab"><font color="Green" title="E92">A1249</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E100">A1268</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E99">A1267</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"/><span class="lab"><font color="Green" title="E103">A1271</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_21"><span class="lab"><font color="Green" title="E92">A1249</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E100">A1268</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E99">A1267</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"/>
<font color="Maroon" title="c29">q2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_21"><span class="lab"><font color="Green" title="E92">A1249</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_21_2"><span class="lab"><font color="Green" title="E97">A1262</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E98">A1266</font></span></a></span>;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A1272</font></span>: 
<font color="Maroon" title="c29">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E101">A1269</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"/><span class="lab"><font color="Green" title="E105">A1273</font></span>: 
<font color="Maroon" title="c29">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_21"><span class="lab"><font color="Green" title="E92">A1249</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_21_2"><span class="lab"><font color="Green" title="E97">A1262</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E98">A1266</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"/><span class="lab"><font color="Green" title="E106">A1274</font></span>: 
( <font color="Maroon" title="c28">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp;  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c29">q2</font>,<font color="Maroon" title="c28">q1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_21_2"><span class="lab"><font color="Green" title="E96">A1261</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_21_2"><span class="lab"><font color="Green" title="E97">A1262</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E12:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"/><span class="lab"><font color="Green" title="E107">A1275</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_21"><span class="lab"><font color="Green" title="E92">A1249</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_21_2"><span class="lab"><font color="Green" title="E97">A1262</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E98">A1266</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1"/>
( <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  or <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E104">A1272</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E106">A1274</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c30">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E108">A1276</font></span>: 
( <font color="Maroon" title="c30">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E109">A1277</font></span>: 
( <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_21_2"><span class="lab"><font color="Green" title="E93">A1258</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_21_2"><span class="lab"><font color="Green" title="E95">A1260</font></span></a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E102">A1270</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E107">A1275</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E109">A1277</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A1278</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E83">A1099</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E101">A1269</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E108">A1276</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E110">A1278</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c28">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E104">A1272</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E106">A1274</font></span></a>, <a class="ref" href="rltopsp1.html#T70" onmouseover="rs('rltopsp1/T70')" onmouseout="rh()">RLTOPSP1:70</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_21_2"><span class="lab"><font color="Green" title="E93">A1258</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_21_2"><span class="lab"><font color="Green" title="E95">A1260</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A1279</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E83">A1099</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E101">A1269</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E103">A1271</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E102">A1270</font></span></a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E108">A1279</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"/>
<font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c27">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p1</font> where <font color="Olive" title="b1">p1</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E104">A1272</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E106">A1274</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c30">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E108">A1280</font></span>: 
( <font color="Maroon" title="c30">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font> &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  )
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E109">A1281</font></span>: 
( <font color="Maroon" title="c28">q1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">q2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_21_2"><span class="lab"><font color="Green" title="E93">A1258</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_21_2"><span class="lab"><font color="Green" title="E95">A1260</font></span></a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Maroon" title="c30">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E103">A1271</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E105">A1273</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E109">A1281</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A1282</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E83">A1099</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"><span class="lab"><font color="Green" title="E101">A1269</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E108">A1280</font></span></a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">j</font>)</span>)</span> where <font color="Olive" title="b1">j</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E110">A1282</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E14:2_1_1_2_7_1_2_1_1_3_21_2_1_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:2_1_1_2_7_1_2_1_1_3_21_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c25">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> or <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:2_1_1_2_7_1_2_1_1_3_21"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font></span><br/>

<a NAME="E5:2_1_1_2_7_1_2_1_1_3_21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A1283</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E1:2_1_1_2"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="ref" href="goboard2.html#T3" onmouseover="rs('goboard2/T3')" onmouseout="rh()">GOBOARD2:3</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_3_1"/>
( <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> or <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_21"><span class="lab"><font color="Green" title="E93">A1283</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_3_1_1"/><span class="lab"><font color="Green" title="E94">A1284</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_21_3_1_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard2.html#T6" onmouseover="rs('goboard2/T6')" onmouseout="rh()">GOBOARD2:6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_2_1_1_3_21_3_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:2_1_1_2"><span class="lab"><font color="Green" title="E36">A44</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_3_1_1"><span class="lab"><font color="Green" title="E94">A1284</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_3_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">k</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">p1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_21_3_1_2"/><span class="lab"><font color="Green" title="E94">A1285</font></span>: 
<font color="Maroon" title="c26">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">x</font>
 <span class="kw">and </span><br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_21_3_1_2"/><span class="lab"><font color="Green" title="E95">A1286</font></span>: 
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">and </span><br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_21_3_1_2"/><span class="lab"><font color="Green" title="E96">A1287</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">and </span><br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_21_3_1_2"/><span class="lab"><font color="Green" title="E97">A1288</font></span>: 
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E83">A1099</font></span></a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> &amp; (  for <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) );<br/><div><span class="lab"><font color="Green" title="E98">A1289</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_3_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]</span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c27">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><span class="kw">take </span><font color="Maroon" title="c28">n</font> = <font color="Maroon" title="c27">n</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c28">n</font>]</span><br/><span class="kw">thus </span><a NAME="E2:2_1_1_2_7_1_2_1_1_3_21_3_1_2_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c28">n</font>]
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_3_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_3_1_2_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">n</font> &amp; <font color="Maroon" title="c28">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21"><span class="lab"><font color="Green" title="E91">A1248</font></span></a>, <a class="ref" href="xreal_1.html#T31" onmouseover="rs('xreal_1/T31')" onmouseout="rh()">XREAL_1:31</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">n</font> holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/>

<a NAME="E2:2_1_1_2_7_1_2_1_1_3_21_3_1_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E3:2_1_1_2_7_1_2_1_1_3_21_3_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A1290</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c29">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c29">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">n</font> implies <font color="Maroon" title="c29">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E4:2_1_1_2_7_1_2_1_1_3_21_3_1_2_1_1"/>
<font color="Maroon" title="c29">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c29">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span><br/>

<span class="kw">then </span><font color="Maroon" title="c29">q</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:2_1_1_2"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_21_3_1_2_1_1"><span class="lab"><font color="Green" title="E99">A1290</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E39:2_1_1_2"><span class="lab"><font color="Green" title="E42">A51</font></span></a>, <a class="txt" href="#E21:2_1_1_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:2_1_1_2_7_1_2_1_1_3_21_3_1_2_1_1"/>
<font color="Maroon" title="c29">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E96">A1287</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E8:2_1_1_2_7_1_2_1_1_3_21_3_1_2"/><span class="lab"><font color="Green" title="E99">A1291</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c27">ma</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:2_1_1_2_7_1_2_1_1_3_21_3_1_2"/><span class="lab"><font color="Green" title="E100">A1292</font></span>: 
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c27">ma</font>] &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">ma</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S6" onmouseover="rs('nat_1/S6')" onmouseout="rh()">NAT_1:sch 6</a>(<a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E99">A1291</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E98">A1289</font></span></a>);<br/></span><span class="kw">reconsider </span><font color="Maroon" title="c28">ma</font> = <font color="Maroon" title="c27">ma</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_3_1_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1"/>
( <font color="Maroon" title="c28">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> or <font color="Maroon" title="c28">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><span class="lab"><font color="Green" title="E101">A1293</font></span>: 
<font color="Maroon" title="c28">ma</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/>
<font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E62">A1011</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A1294</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E4:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21"><span class="lab"><font color="Green" title="E91">A1248</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E101">A1293</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A1295</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span>
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">m1</font> = <font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" onmouseover="rs('int_1/T5')" onmouseout="rh()">INT_1:5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c30">q</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">m1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E9:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><span class="lab"><font color="Green" title="E104">A1296</font></span>: 
<font color="Maroon" title="c28">ma</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E101">A1293</font></span></a>, <a class="ref" href="xreal_1.html#T43" onmouseover="rs('xreal_1/T43')" onmouseout="rh()">XREAL_1:43</a></span>;<br/><a NAME="E10:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A1297</font></span>: 
<font color="Maroon" title="c30">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_21"><span class="lab"><font color="Green" title="E92">A1249</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E103">A1295</font></span></a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><span class="lab"><font color="Green" title="E106">A1298</font></span>: 
<font color="Maroon" title="c30">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_21"><span class="lab"><font color="Green" title="E92">A1249</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E104">A1296</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E103">A1295</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c31">lq</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">e</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c30">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">e</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/><a NAME="E12:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><span class="lab"><font color="Green" title="E107">A1299</font></span>: 
<font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j1</font>
 ;<br/><a NAME="E13:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><span class="lab"><font color="Green" title="E108">A1300</font></span>: 
<font color="Maroon" title="c19">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E102">A1294</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A1301</font></span>: <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">ma</font> = 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21"><span class="lab"><font color="Green" title="E91">A1248</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E101">A1293</font></span></a>, <a class="ref" href="finseq_4.html#T69" onmouseover="rs('finseq_4/T69')" onmouseout="rh()">FINSEQ_4:69</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E108">A1300</font></span></a>, <a class="txt" href="#E12:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E107">A1299</font></span></a></span>
;<br/><a NAME="E15:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E100">A1292</font></span></a></span>;<br/><a NAME="E16:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A1302</font></span>: 
<font color="Maroon" title="c26">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E97">A1288</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E17:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E18:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">A1303</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E103">A1295</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E19:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><span class="lab"><font color="Green" title="E112">A1304</font></span>: 
<font color="Maroon" title="c29">m1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">ma</font>
 ;<br/><a NAME="E20:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c30">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">q</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> &amp;  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c30">q</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E101">A1293</font></span></a>, <a class="txt" href="#E14:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E109">A1301</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E111">A1303</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>;<br/><a NAME="E21:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">e</font> where <font color="Olive" title="b1">e</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">e</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c30">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">e</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">e</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c6">j1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E80">A1083</font></span></a>, <a class="txt" href="#E33:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E82">A1098</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E105">A1297</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E106">A1298</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a></span>;<br/><a NAME="E22:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A1305</font></span>: 
<font color="Maroon" title="c26">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E95">A1286</font></span></a>, <a class="txt" href="#E16:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E110">A1302</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E106">A1298</font></span></a></span>;<br/><a NAME="E23:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c29">m1</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E101">A1293</font></span></a>, <a class="txt" href="#E18:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E111">A1303</font></span></a>, <a class="txt" href="#E19:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E112">A1304</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E24:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E94">A1285</font></span></a>, <a class="txt" href="#E22:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_1"><span class="lab"><font color="Green" title="E113">A1305</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/>
<font color="Maroon" title="c28">ma</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font></span><br/><div class="add"><a NAME="E2:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c28">ma</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E100">A1292</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A1306</font></span>: 
<font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c29">qa</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c28">ma</font>, <font color="Maroon" title="c30">qa1</font> = <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><span class="kw">set </span><font color="Maroon" title="c31">lma</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> ;<br/><a NAME="E5:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/><span class="lab"><font color="Green" title="E102">A1307</font></span>: 
<font color="Maroon" title="c30">qa1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><a NAME="E6:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/><span class="lab"><font color="Green" title="E103">A1308</font></span>: 
<font color="Maroon" title="c29">qa</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E100">A1292</font></span></a></span>;<br/><a NAME="E7:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/><span class="lab"><font color="Green" title="E104">A1309</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E100">A1292</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E8:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A1310</font></span>: 
<font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_21"><span class="lab"><font color="Green" title="E92">A1249</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E101">A1306</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E106">A1311</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2_1"/>
<font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">q</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b1">q</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">p1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a> 
 ;<br/><a NAME="E3:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c28">ma</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E100">A1292</font></span></a>, <a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E101">A1306</font></span></a>, <a class="txt" href="#E7:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E104">A1309</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" onmouseover="rs('xreal_1/T29')" onmouseout="rh()">XREAL_1:29</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E10:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/><span class="lab"><font color="Green" title="E107">A1312</font></span>: 
( <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c29">qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">qa</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c29">qa</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a></span>)</span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1_2_7_1_2_1_1_3_21"><span class="lab"><font color="Green" title="E92">A1249</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E100">A1292</font></span></a>, <a class="ref" href="euclid.html#T53" onmouseover="rs('euclid/T53')" onmouseout="rh()">EUCLID:53</a></span>;<br/><a NAME="E11:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/><span class="lab"><font color="Green" title="E108">A1313</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">ma</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E100">A1292</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><span class="kw">then </span> <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font>) = 
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<font color="Maroon" title="c29">qa</font>,<font color="Maroon" title="c30">qa1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E101">A1306</font></span></a>, <a class="ref" href="topreal1.html#D3" onmouseover="rs('topreal1/D3')" onmouseout="rh()">TOPREAL1:def 3</a></span>
<br/>.= 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">p2</font> where <font color="Olive" title="b1">p2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : ( <font color="Olive" title="b1">p2</font> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">i2</font>)</span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">`2</a>  &amp; <font color="Maroon" title="c29">qa</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c30">qa1</font> <a href="euclid.html#K17" title="EUCLID:func.17">`1</a>  ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E103">A1308</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E106">A1311</font></span></a>, <a class="txt" href="#E8:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E105">A1310</font></span></a>, <a class="txt" href="#E10:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E107">A1312</font></span></a>, <a class="txt" href="#E5:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E102">A1307</font></span></a>, <a class="ref" href="topreal3.html#T10" onmouseover="rs('topreal3/T10')" onmouseout="rh()">TOPREAL3:10</a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>
;<br/><a NAME="E13:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">A1314</font></span>: 
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E94">A1285</font></span></a>, <a class="txt" href="#E4:2_1_1_2_7_1_2_1_1_3_21_3_1_2"><span class="lab"><font color="Green" title="E95">A1286</font></span></a>, <a class="txt" href="#E6:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E103">A1308</font></span></a>, <a class="txt" href="#E9:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E106">A1311</font></span></a></span>;<br/><a NAME="E14:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c28">ma</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c21">g</font>,<font color="Olive" title="b1">i</font>)</span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> ) </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E101">A1306</font></span></a>, <a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E108">A1313</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:2_1_1_2_7_1_2_1_1_3_21_3_1_2_2_1_2"><span class="lab"><font color="Green" title="E109">A1314</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:2_1_1_2_7_1_2_1_1_3_21_3_1_2"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:2_1_1_2_7_1_2_1_1_3_21"/>
<font color="Maroon" title="c25">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c21">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E51:2_1_1_2_7_1_2_1_1_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:2_1_1_2"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/><a NAME="E52:2_1_1_2_7_1_2_1_1_3"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:2_1_1_2"><span class="lab"><font color="Green" title="E37">A45</font></span></a>, <a class="ref" href="finseq_4.html#T68" onmouseover="rs('finseq_4/T68')" onmouseout="rh()">FINSEQ_4:68</a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:2_1_1_2"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E16:2_1_1_2"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="finseq_4.html#T71" onmouseover="rs('finseq_4/T71')" onmouseout="rh()">FINSEQ_4:71</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> )</span><br/><a NAME="E54:2_1_1_2_7_1_2_1_1_3"/><span class="lab"><font color="Green" title="E91">A1315</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">g1</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>;<br/><a NAME="E55:2_1_1_2_7_1_2_1_1_3"/>
<font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E62">A1011</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E56:2_1_1_2_7_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A1316</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" onmouseover="rs('xreal_1/T19')" onmouseout="rh()">XREAL_1:19</a></span>;<br/><a NAME="E57:2_1_1_2_7_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A1317</font></span>: 
<font color="Maroon" title="c19">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c21">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span>)</span> = 
<font color="Maroon" title="c20">g2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E91">A1315</font></span></a>, <a class="ref" href="finseq_4.html#T69" onmouseover="rs('finseq_4/T69')" onmouseout="rh()">FINSEQ_4:69</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c19">l</font></span>)</span>,<font color="Maroon" title="c9">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E68">A1017</font></span></a>, <a class="txt" href="#E57:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E93">A1317</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E12:2_1_1_2"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:2_1_1_2_7_1_2"><span class="lab"><font color="Green" title="E61">A698</font></span></a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font></span><br/><span class="kw">thus </span><a NAME="E59:2_1_1_2_7_1_2_1_1_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E2:2_1_1_2"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E36:2_1_1_2"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="txt" href="#E56:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E92">A1316</font></span></a>, <a class="txt" href="#E54:2_1_1_2_7_1_2_1_1_3"><span class="lab"><font color="Green" title="E91">A1315</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:2_1_1_2_7_1_2"/>
 ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E59:2_1_1_2"/>
 ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E2:2"/><span class="lab"><font color="Green" title="E4">A1318</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>) ) ) &amp; <font color="Maroon" title="c3">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  implies  ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_2"/><span class="lab"><font color="Green" title="E5">A1319</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:2_2"/><span class="lab"><font color="Green" title="E6">A1320</font></span>: 
( (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>) ) ) &amp; <font color="Maroon" title="c3">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">f</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c4">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c4">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c4">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g</font> )</span><br/>

<a NAME="E3:2_2"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_2"><span class="lab"><font color="Green" title="E5">A1319</font></span></a></span>;<br/>
<a NAME="E4:2_2"/><span class="kw">then </span>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> holds <br/> for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font>) &amp; <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font>) holds <br/><span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">i</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1"><a href="complex1.html#K9" title="COMPLEX1:func.9">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">j</font></span>)</span></span><a href="complex1.html#K9" title="COMPLEX1:func.9">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
;<br/>
<span class="kw">hence </span><a NAME="E5:2_2"/>
( <font color="Maroon" title="c4">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c4">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c4">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c4">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_2"><span class="lab"><font color="Green" title="E6">A1320</font></span></a>, <a class="ref" href="goboard1.html#D9" onmouseover="rs('goboard1/D9')" onmouseout="rh()">GOBOARD1:def 9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:2"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E2:2"><span class="lab"><font color="Green" title="E4">A1318</font></span></a>, <a class="txt" href="#E1:2"><span class="lab"><font color="Green" title="E3">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:2"/>
( (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>) ) ) &amp; <font color="Maroon" title="c1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Maroon" title="c1">f</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  implies  ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V3" title="TOPREAL1:attr.3">s.n.c.</a>  &amp; <font color="Olive" title="b1">g</font> is  <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>  &amp;  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 &amp; <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">g</font> ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
