<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">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" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies  for <font color="Olive" title="b1">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 holds <br/> for <font color="Olive" title="b2">E</font> being    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Olive" title="b1">A</font>  st <font color="Olive" title="b2">E</font> <a href="funct_1.html#K1" title="FUNCT_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="Olive" title="b2">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> holds <br/> for <font color="Olive" title="b3">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">k</font></span>)</span><br/>  for <font color="Olive" title="b4">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b5">pn</font> where <font color="Olive" title="b5">pn</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> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b5">pn</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Olive" title="b2">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">P</font> </span> } </span>  holds <br/>( <font color="Olive" title="b3">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Olive" title="b4">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:38"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">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">  for <font color="Olive" title="b1">A</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 holds <br/> for <font color="Olive" title="b2">E</font> being    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Olive" title="b1">A</font>  st <font color="Olive" title="b2">E</font> <a href="funct_1.html#K1" title="FUNCT_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="Olive" title="b2">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> holds <br/> for <font color="Olive" title="b3">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">k</font></span>)</span><br/>  for <font color="Olive" title="b4">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b5">pn</font> where <font color="Olive" title="b5">pn</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> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b5">pn</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Olive" title="b2">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">P</font> </span> } </span>  holds <br/>( <font color="Olive" title="b3">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Olive" title="b4">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">V</font> = <font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">TRn</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</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"> (  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 implies  for <font color="Olive" title="b1">E</font> being    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c5">A</font>  st <font color="Olive" title="b1">E</font> <a href="funct_1.html#K1" title="FUNCT_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="Olive" title="b1">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> holds <br/> for <font color="Olive" title="b2">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">k</font></span>)</span><br/>  for <font color="Olive" title="b3">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b4">pn</font> where <font color="Olive" title="b4">pn</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> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b4">pn</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Olive" title="b1">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">P</font> </span> } </span>  holds <br/>( <font color="Olive" title="b2">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Olive" title="b3">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">A1</font> = <font color="Maroon" title="c5">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="rlaffin3.html#E10"><span class="lab"><font color="Green" title="E10">Lm3</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">TRk</font> =  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">k</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">cA</font> = <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1;<br/>
<span class="kw">assume </span><a NAME="E3:38"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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">  for <font color="Olive" title="b1">E</font> being    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c5">A</font>  st <font color="Olive" title="b1">E</font> <a href="funct_1.html#K1" title="FUNCT_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="Olive" title="b1">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> holds <br/> for <font color="Olive" title="b2">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">k</font></span>)</span><br/>  for <font color="Olive" title="b3">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b4">pn</font> where <font color="Olive" title="b4">pn</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> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b4">pn</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Olive" title="b1">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">P</font> </span> } </span>  holds <br/>( <font color="Olive" title="b2">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Olive" title="b3">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<a NAME="E4:38"/>
 <a href="rlvect_5.html#K1" title="RLVECT_5:func.1">dim</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#T4" target="_self" onmouseover="rs('rlaffin3/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E5:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A3</font></span>: 
 <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="rlaffin3.html#T6" target="_self" onmouseover="rs('rlaffin3/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E6:38"/>
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T66" onmouseover="rs('euclid/T66')" onmouseout="rh()">EUCLID:66</a></span>;<br/>
<a NAME="E7:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A4</font></span>: 
 <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="rlaffin3.html#E4"><span class="lab"><font color="Green" title="E4">Lm2</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">E</font> be    <a href="rlaffin3.html#M2" title="RLAFFIN3:mode.2">Enumeration</a> of <font color="Maroon" title="c5">A</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">E</font> <a href="funct_1.html#K1" title="FUNCT_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="c9">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> implies  for <font color="Olive" title="b1">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">k</font></span>)</span><br/>  for <font color="Olive" title="b2">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">pn</font> where <font color="Olive" title="b3">pn</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> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b3">pn</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">P</font> </span> } </span>  holds <br/>( <font color="Olive" title="b1">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Olive" title="b2">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">e</font> = <font color="Maroon" title="c9">E</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="prvect_1.html#K6" title="PRVECT_1:func.6">-VectSp_over</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="rlaffin3.html#E10"><span class="lab"><font color="Green" title="E10">Lm3</font></span></a></span>;<br/>
<a NAME="E9:38"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#D1" target="_self" onmouseover="rs('rlaffin3/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E10:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T62" onmouseover="rs('finseq_4/T62')" onmouseout="rh()">FINSEQ_4:62</a></span>;<br/>
<a NAME="E11:38"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</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#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E12:38"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix13.html#K10" title="MATRIX13:func.10">FinS2MX</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">e</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E10:38"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">M</font> =  <a href="matrix13.html#K10" title="MATRIX13:func.10">FinS2MX</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">e</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font></span>)</span> as   <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of <font color="Maroon" title="c1">n</font>,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a> ;<br/>
<a NAME="E14:38"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a>, <a class="ref" href="xreal_1.html#T233" onmouseover="rs('xreal_1/T233')" onmouseout="rh()">XREAL_1:233</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">MT</font> =  <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">M</font>;<br/>
<span class="kw">assume </span><a NAME="E15:38"/><span class="lab"><font color="Green" title="E37">A8</font></span>: 
<font color="Maroon" title="c9">E</font> <a href="funct_1.html#K1" title="FUNCT_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="c9">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">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">  for <font color="Olive" title="b1">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">k</font></span>)</span><br/>  for <font color="Olive" title="b2">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">pn</font> where <font color="Olive" title="b3">pn</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> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b3">pn</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">P</font> </span> } </span>  holds <br/>( <font color="Olive" title="b1">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Olive" title="b2">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c13">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">pn</font> where <font color="Olive" title="b2">pn</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> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b2">pn</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">P</font> </span> } </span>  holds <br/>( <font color="Maroon" title="c13">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Olive" title="b1">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c14">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</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="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">pn</font> where <font color="Olive" title="b1">pn</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> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">pn</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">P</font> </span> } </span>  implies ( <font color="Maroon" title="c13">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Maroon" title="c14">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  ) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c15">PP</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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Olive" title="b1">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">P</font> </span> } </span> ;<br/>
<a NAME="E16:38"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Olive" title="b1">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">P</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="38_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">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="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Olive" title="b1">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">P</font> </span> } </span>  or <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:38_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Olive" title="b1">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">P</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="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><br/>

<a NAME="E2:38_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> &amp; <font color="Olive" title="b1">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">P</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:38_1"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">PP</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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Olive" title="b1">v</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">P</font> </span> } </span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<span class="kw">assume </span><a NAME="E18:38"/><span class="lab"><font color="Green" title="E38">A9</font></span>: 
<font color="Maroon" title="c14">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">P</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="c13">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Maroon" title="c14">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<a NAME="E19:38"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a></span>;<br/>
<a NAME="E20:38"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">E</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:38"><span class="lab"><font color="Green" title="E35">A6</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="E21:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A10</font></span>: 
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E9:38"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E22:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A11</font></span>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">A1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">E</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="rlaffin3.html#T23" target="_self" onmouseover="rs('rlaffin3/T23')" onmouseout="rh()">Th23</a>, <a class="ref" href="matrtop2.html#T6" onmouseover="rs('matrtop2/T6')" onmouseout="rh()">MATRTOP2:6</a></span>;<br/>
<a NAME="E23:38"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="rlvect_3.html#K1" title="RLVECT_3:func.1">Lin</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrtop2.html#T6" onmouseover="rs('matrtop2/T6')" onmouseout="rh()">MATRTOP2:6</a></span>;<br/>
<a NAME="E24:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A12</font></span>: 
 <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">lines</a> <font color="Maroon" title="c11">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <font color="Maroon" title="c6">A1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:38"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E7:38"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E22:38"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="vectsp_4.html#T29" onmouseover="rs('vectsp_4/T29')" onmouseout="rh()">VECTSP_4:29</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">BE</font> = <font color="Maroon" title="c9">E</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> as    <a href="matrlin.html#M1" title="MATRLIN:mode.1">OrdBasis</a> of  <a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p1">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">lines</a> <font color="Maroon" title="c11">M</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E21:38"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="ref" href="rlaffin3.html#T23" target="_self" onmouseover="rs('rlaffin3/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<a NAME="E26:38"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">BE</font> is   <a href="vectsp_7.html#NM1" title="VECTSP_7:NM.1">Basis</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">lines</a> <font color="Maroon" title="c11">M</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D2" onmouseover="rs('matrlin/D2')" onmouseout="rh()">MATRLIN:def 2</a></span>;<br/>
<a NAME="E27:38"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">BE</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_7.html#D3" onmouseover="rs('vectsp_7/D3')" onmouseout="rh()">VECTSP_7:def 3</a></span>;<br/>
<a NAME="E28:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A13</font></span>: 
 <a href="matrix13.html#K9" title="MATRIX13:func.9">lines</a> <font color="Maroon" title="c11">M</font> is  <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:38"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="vectsp_9.html#T11" onmouseover="rs('vectsp_9/T11')" onmouseout="rh()">VECTSP_9:11</a></span>;<br/>
<a NAME="E29:38"/>
<font color="Maroon" title="c17">BE</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:38"><span class="lab"><font color="Green" title="E36">A7</font></span></a></span>;<br/>
<a NAME="E30:38"/><span class="kw">then </span>
<font color="Maroon" title="c11">M</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="matrlin.html#D2" onmouseover="rs('matrlin/D2')" onmouseout="rh()">MATRLIN:def 2</a></span>;<br/>
<a NAME="E31:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A14</font></span>: 
 <a href="matrix13.html#K8" title="MATRIX13:func.8">the_rank_of</a> <font color="Maroon" title="c11">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:38"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="matrix13.html#T121" onmouseover="rs('matrix13/T121')" onmouseout="rh()">MATRIX13:121</a></span>;<br/>
<a NAME="E32:38"/><span class="kw">then </span>
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Maroon" title="c11">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix13.html#T83" onmouseover="rs('matrix13/T83')" onmouseout="rh()">MATRIX13:83</a></span>;<br/>
<a NAME="E33:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A15</font></span>: 
<font color="Maroon" title="c11">M</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="laplace.html#T34" onmouseover="rs('laplace/T34')" onmouseout="rh()">LAPLACE:34</a></span>;<br/>
<a NAME="E34:38"/>
 <a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">M</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:38"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="ref" href="matrtop1.html#T41" onmouseover="rs('matrtop1/T41')" onmouseout="rh()">MATRTOP1:41</a></span>;<br/>
<a NAME="E35:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A16</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E36:38"/><span class="lab"><font color="Green" title="E46">A17</font></span>: 
<font color="Maroon" title="c14">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">M</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c16">PP</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="38_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">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="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">M</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c16">PP</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:38_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</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="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">M</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c16">PP</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:38_2"/><span class="lab"><font color="Green" title="E47">A18</font></span>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">v</font>
 <span class="kw">and </span><br/><a NAME="E4:38_2"/><span class="lab"><font color="Green" title="E48">A19</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:38"><span class="lab"><font color="Green" title="E38">A9</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:38_2"/><span class="lab"><font color="Green" title="E49">A20</font></span>: 
( <font color="Maroon" title="c20">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">M</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:38"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E7:38_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="rlaffin3.html#T16" target="_self" onmouseover="rs('rlaffin3/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<a NAME="E8:38_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><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/>
<a NAME="E9:38_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T92" onmouseover="rs('finseq_2/T92')" onmouseout="rh()">FINSEQ_2:92</a></span>;<br/>
<a NAME="E10:38_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>
 
;<br/>
<a NAME="E11:38_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D1" onmouseover="rs('euclid/D1')" onmouseout="rh()">EUCLID:def 1</a></span>;<br/>
<a NAME="E12:38_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A21</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T22" onmouseover="rs('euclid/T22')" onmouseout="rh()">EUCLID:22</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">w</font> = <font color="Maroon" title="c19">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">lines</a> <font color="Maroon" title="c11">M</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:38"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E5:38"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E21:38"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E7:38"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E22:38"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="rlaffin3.html#T11" target="_self" onmouseover="rs('rlaffin3/T11')" onmouseout="rh()">Th11</a></span>;<br/>
<a NAME="E14:38_2"/><span class="lab"><font color="Green" title="E51">A22</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">v</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="finseq_1.html#T82" onmouseover="rs('finseq_1/T82')" onmouseout="rh()">FINSEQ_1:82</a></span>;<br/>
<font color="Maroon" title="c20">f</font> = 
<font color="Maroon" title="c21">w</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c17">BE</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:38"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E6:38_2"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="ref" href="matrtop2.html#T17" onmouseover="rs('matrtop2/T17')" onmouseout="rh()">MATRTOP2:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">w</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E21:38"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E24:38"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="ref" href="rlaffin3.html#T24" target="_self" onmouseover="rs('rlaffin3/T24')" onmouseout="rh()">Th24</a></span>
;<br/>
<a NAME="E16:38_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">PP</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:38"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E4:38_2"><span class="lab"><font color="Green" title="E48">A19</font></span></a>, <a class="txt" href="#E12:38_2"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="txt" href="#E14:38_2"><span class="lab"><font color="Green" title="E51">A22</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:38_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">M</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c16">PP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="txt" href="#E6:38_2"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</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="E37:38"/>
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">M</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c16">PP</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">B</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="38_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">y</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="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">M</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c16">PP</font> or <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:38_3"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">M</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c16">PP</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="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:38_3"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">M</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:38_3"/><span class="lab"><font color="Green" title="E47">A23</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">PP</font>
 <span class="kw">and </span><br/><a NAME="E5:38_3"/><span class="lab"><font color="Green" title="E48">A24</font></span>: 
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">f</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E7:38_3"/><span class="lab"><font color="Green" title="E49">A25</font></span>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">f</font> &amp; <font color="Maroon" title="c20">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c2">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_3"><span class="lab"><font color="Green" title="E47">A23</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">MTf</font> = <span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">M</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">f</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp_7.html#K1" title="VECTSP_7:func.1">Lin</a> <span class="p2">(<span class="default"><a href="matrix13.html#K9" title="MATRIX13:func.9">lines</a> <font color="Maroon" title="c11">M</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:38"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E5:38"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E21:38"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E7:38"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E22:38"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="rlaffin3.html#T11" target="_self" onmouseover="rs('rlaffin3/T11')" onmouseout="rh()">Th11</a></span>;<br/>
<font color="Maroon" title="c20">f</font> = 
<font color="Maroon" title="c21">MTf</font> <a href="matrlin.html#K9" title="MATRLIN:func.9">|--</a> <font color="Maroon" title="c17">BE</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:38"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="matrtop2.html#T17" onmouseover="rs('matrtop2/T17')" onmouseout="rh()">MATRTOP2:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">MTf</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:38"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E21:38"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E24:38"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="ref" href="rlaffin3.html#T24" target="_self" onmouseover="rs('rlaffin3/T24')" onmouseout="rh()">Th24</a></span>
;<br/>
<a NAME="E10:38_3"/><span class="kw">then </span>
<font color="Maroon" title="c20">f</font> <a href="finseq_1.html#K16" title="FINSEQ_1:func.16">|</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">MTf</font> <a href="rlaffin3.html#K1" title="RLAFFIN3:func.1">|--</a> <font color="Maroon" title="c9">E</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c2">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E14:38"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="finseq_1.html#T82" onmouseover="rs('finseq_1/T82')" onmouseout="rh()">FINSEQ_1:82</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:38_3"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:38"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E5:38_3"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E7:38_3"><span class="lab"><font color="Green" title="E49">A25</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="E38:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A26</font></span>: 
<span class="p1">(<span class="default"><a href="matrtop1.html#K3" title="MATRTOP1:func.3">Mx2Tran</a> <font color="Maroon" title="c11">M</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c16">PP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:38"><span class="lab"><font color="Green" title="E46">A17</font></span></a></span>;<br/>
<a NAME="E39:38"/>
( <font color="Maroon" title="c13">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Maroon" title="c16">PP</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="rlaffin3.html#T7" target="_self" onmouseover="rs('rlaffin3/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E40:38"/>
( <font color="Maroon" title="c13">P</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  iff <font color="Maroon" title="c14">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:38"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="txt" href="#E38:38"><span class="lab"><font color="Green" title="E47">A26</font></span></a>, <a class="ref" href="topgrp_1.html#T25" onmouseover="rs('topgrp_1/T25')" onmouseout="rh()">TOPGRP_1:25</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"> verum</span><br/>


</div>
