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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K15" title="POLNOT_1:func.15">-head</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="polnot_1.html#K16" title="POLNOT_1:func.16">-tail</a> <font color="Maroon" title="c3">p</font></span>)</span> );<br/>
<a NAME="E1:126_1_1"/><span class="lab"><font color="Green" title="E70">A1</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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="c2">n</font> holds <br/> ex <font color="Olive" title="b2">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">s</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="126_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</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="c4">i</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="c2">n</font> implies  ex <font color="Olive" title="b1">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>,<font color="Olive" title="b1">s</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:126_1_1_1"/>
<font color="Maroon" title="c4">i</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="c2">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">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>,<font color="Olive" title="b1">s</font>]</span><br/>

<a NAME="E2:126_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A3</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</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/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:126_1_1_1"/><span class="lab"><font color="Green" title="E72">A4</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T6" onmouseover="rs('nat_1/T6')" onmouseout="rh()">NAT_1:6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">l</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:126_1_1_1"/><span class="lab"><font color="Green" title="E73">A5</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:126_1_1_1"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="ref" href="nat_1.html#T10" onmouseover="rs('nat_1/T10')" onmouseout="rh()">NAT_1:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c3">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">l</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:126_1_1_1"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="txt" href="#E6:126_1_1_1"><span class="lab"><font color="Green" title="E73">A5</font></span></a></span>;<br/>
<span class="kw">take </span>
 <a href="polnot_1.html#K27" title="POLNOT_1:func.27">decomp</a> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c7">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">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>, <a href="polnot_1.html#K27" title="POLNOT_1:func.27">decomp</a> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c7">p</font>)]</span><br/>

<span class="kw">thus </span><a NAME="E8:126_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>, <a href="polnot_1.html#K27" title="POLNOT_1:func.27">decomp</a> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c7">p</font>)]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:126_1_1_1"><span class="lab"><font color="Green" title="E72">A4</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>
<span class="kw">consider </span><font color="Maroon" title="c4">q</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E3:126_1_1"/><span class="lab"><font color="Green" title="E71">A10</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">q</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="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E4:126_1_1"/><span class="lab"><font color="Green" title="E72">A11</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</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="c2">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S5" onmouseover="rs('finseq_1/S5')" onmouseout="rh()">FINSEQ_1:sch 5</a>(<a class="txt" href="#E1:126_1_1"><span class="lab"><font color="Green" title="E70">A1</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c4">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"> (  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">q</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="c2">n</font> &amp; (  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="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="c2">n</font> holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K15" title="POLNOT_1:func.15">-head</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="polnot_1.html#K16" title="POLNOT_1:func.16">-tail</a> <font color="Maroon" title="c3">p</font></span>)</span> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:126_1_1"/>
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">q</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="c2">n</font> &amp; (  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="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="c2">n</font> holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K15" title="POLNOT_1:func.15">-head</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="polnot_1.html#K4" title="POLNOT_1:func.4">^^</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="polnot_1.html#K16" title="POLNOT_1:func.16">-tail</a> <font color="Maroon" title="c3">p</font></span>)</span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:126_1_1"><span class="lab"><font color="Green" title="E71">A10</font></span></a>, <a class="txt" href="#E4:126_1_1"><span class="lab"><font color="Green" title="E72">A11</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/>


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