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

<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">M</font> be  <a href="matrix_1.html#V2" title="MATRIX_1:attr.2">diagonal</a> <a href="matrix_1.html#NM4" title="MATRIX_1:NM.4">Matrix</a> of <font color="Maroon" title="c1">K</font>;<br/><b>let </b><font color="Maroon" title="c3">NonZero</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:116"/><i><font color="Green" title="E99">A1</font></i>: 
<font color="Maroon" title="c3">NonZero</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b1">i</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b1">i</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> <font color="Maroon" title="c1">K</font> ) </span>}</span> 
 ;<br/>

<b>let </b><font color="Maroon" title="c4">P</font>, <font color="Maroon" title="c5">Q</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:116"/><i><font color="Green" title="E100">A2</font></i>: 
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">M</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Q</font> )
 <b>and </b><br/><a NAME="E3:116"/><i><font color="Green" title="E101">A3</font></i>: 
 <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span> <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> <font color="Maroon" title="c1">K</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">ES</font> =  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>;<br/>
<b>set </b><font color="Maroon" title="c7">S</font> =  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>;<br/>
<b>set </b><font color="Maroon" title="c8">SP</font> =  <a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">P</font>;<br/>
<b>set </b><font color="Maroon" title="c9">SQ</font> =  <a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">Q</font>;<br/>
<a NAME="E4:116"/><i><font color="Green" title="E102">A4</font></i>: 
 <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font>
 
<b>by </b><i><a class="txt" href="matrix13.html#E2:116"><i><font color="Green" title="E100">A2</font></i></a>, <a class="ref" href="matrix13.html#D3" target="_self" title="MATRIX13:def.3">Def3</a></i>;<br/>
<a NAME="E5:116"/><i><font color="Green" title="E103">A5</font></i>: 
(  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span> <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#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span></span>)</span></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_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p4">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">P</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</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="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span></span>)</span> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">P</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E2:116"><i><font color="Green" title="E100">A2</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<b>thus </b><a NAME="E6:116"/>
<font color="Maroon" title="c4">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">NonZero</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:116_1"/>
not <font color="Maroon" title="c4">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">NonZero</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:116_1"/><i><font color="Green" title="E104">A6</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> &amp; not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">NonZero</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:116_1"/>
( <font color="Maroon" title="c4">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">M</font></span>)</span> &amp; <font color="Maroon" title="c5">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">M</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E2:116"><i><font color="Green" title="E100">A2</font></i></a>, <a class="ref" href="matrix13.html#T67" target="_self" title="MATRIX13:th.67">Th67</a></i>;<br/>
<a NAME="E5:116_1"/><b>then </b><i><font color="Green" title="E105">A7</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">P</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">P</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a>, <a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:116_1"/><i><font color="Green" title="E106">A8</font></i>: 
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">P</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E3:116_1"><i><font color="Green" title="E104">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c11">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="matrix13.html#E3:116_1"><i><font color="Green" title="E104">A6</font></i></a>, <a class="txt" href="matrix13.html#E7:116_1"><i><font color="Green" title="E106">A8</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">L</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c13">y</font>;<br/>
<a NAME="E9:116_1"/><i><font color="Green" title="E107">A9</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c13">y</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c13">y</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="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c13">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:116_1_1"/><i><font color="Green" title="E108">A10</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> &amp; <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span> )
 ;<br/><a NAME="E2:116_1_1"/>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:116_1_1"/><b>then </b><i><font color="Green" title="E109">A11</font></i>: 
( <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN: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_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN: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> <span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E2:116"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="matrix13.html#E5:116_1"><i><font color="Green" title="E105">A7</font></i></a>, <a class="txt" href="matrix13.html#E7:116_1"><i><font color="Green" title="E106">A8</font></i></a>, <a class="txt" href="matrix13.html#E1:116_1_1"><i><font color="Green" title="E108">A10</font></i></a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/><a NAME="E4:116_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c15">i</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_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E5:116"><i><font color="Green" title="E103">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E5:116_1_1"/><b>then </b><i><font color="Green" title="E110">A12</font></i>: 
( <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</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_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">M</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c13">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c15">i</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E2:116"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="matrix13.html#E5:116_1"><i><font color="Green" title="E105">A7</font></i></a>, <a class="txt" href="matrix13.html#E7:116_1"><i><font color="Green" title="E106">A8</font></i></a>, <a class="txt" href="matrix13.html#E3:116_1_1"><i><font color="Green" title="E109">A11</font></i></a>, <a class="ref" href="matrix13.html#D1" target="_self" title="MATRIX13:def.1">Def1</a>, <a class="ref" href="matrix13.html#T17" target="_self" title="MATRIX13:th.17">Th17</a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/><a NAME="E6:116_1_1"/>
( <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">x</font> or <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> )
 ;<br/><a NAME="E7:116_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c13">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:116"><i><font color="Green" title="E99">A1</font></i></a>, <a class="txt" href="matrix13.html#E3:116_1"><i><font color="Green" title="E104">A6</font></i></a>, <a class="txt" href="matrix13.html#E5:116_1_1"><i><font color="Green" title="E110">A12</font></i></a>, <a class="ref" href="matrix_1.html#D15" title="MATRIX_1:def.15">MATRIX_1:def 15</a>, <a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>;<br/><b>hence </b><a NAME="E8:116_1_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c13">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c13">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E3:116_1_1"><i><font color="Green" title="E109">A11</font></i></a>, <a class="ref" href="fvsum_1.html#T63" title="FVSUM_1:th.63">FVSUM_1:63</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:116_1"/><b>then </b>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c13">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix13.html#E9:116_1"><i><font color="Green" title="E107">A9</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>
<a NAME="E12:116_1"/><b>then </b>
(  <a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p1">(<span class="default"><a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c13">y</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c13">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span></span>)</span> &amp;  <a href="matrix11.html#NK5" title="MATRIX11:NK.5">RLine</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c13">y</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c13">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E4:116"><i><font color="Green" title="E102">A4</font></i></a>, <a class="txt" href="matrix13.html#E5:116_1"><i><font color="Green" title="E105">A7</font></i></a>, <a class="txt" href="matrix13.html#E7:116_1"><i><font color="Green" title="E106">A8</font></i></a>, <a class="ref" href="matrix11.html#T30" title="MATRIX11:th.30">MATRIX11:30</a>, <a class="ref" href="matrix11.html#T35" title="MATRIX11:th.35">MATRIX11:35</a></i>;<br/>
<b>hence </b><a NAME="E13:116_1"/>
contradiction
 <b>by </b><i><a class="txt" href="matrix13.html#E3:116"><i><font color="Green" title="E101">A3</font></i></a>, <a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:116"/>
<font color="Maroon" title="c5">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">NonZero</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:116_2"/>
not <font color="Maroon" title="c5">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">NonZero</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:116_2"/><i><font color="Green" title="E104">A13</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> &amp; not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">NonZero</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:116_2"/>
( <font color="Maroon" title="c4">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">M</font></span>)</span> &amp; <font color="Maroon" title="c5">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">M</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E2:116"><i><font color="Green" title="E100">A2</font></i></a>, <a class="ref" href="matrix13.html#T67" target="_self" title="MATRIX13:th.67">Th67</a></i>;<br/>
<a NAME="E5:116_2"/><b>then </b><i><font color="Green" title="E105">A14</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">Q</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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">Q</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a>, <a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:116_2"/><i><font color="Green" title="E106">A15</font></i>: 
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">Q</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E3:116_2"><i><font color="Green" title="E104">A13</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c11">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="matrix13.html#E3:116_2"><i><font color="Green" title="E104">A13</font></i></a>, <a class="txt" href="matrix13.html#E7:116_2"><i><font color="Green" title="E106">A15</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">C</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p1">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c13">y</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:116_2_1"/><i><font color="Green" title="E107">A16</font></i>: 
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R2" title="HIDDEN: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="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">P</font></span>)</span>
 ;<br/><a NAME="E2:116_2_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c13">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E2:116"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="matrix13.html#E5:116"><i><font color="Green" title="E103">A5</font></i></a>, <a class="txt" href="matrix13.html#E5:116_2"><i><font color="Green" title="E105">A14</font></i></a>, <a class="txt" href="matrix13.html#E7:116_2"><i><font color="Green" title="E106">A15</font></i></a>, <a class="txt" href="matrix13.html#E1:116_2_1"><i><font color="Green" title="E107">A16</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:116_2_1"/><b>then </b><i><font color="Green" title="E108">A17</font></i>: 
( <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c12">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font></span>)</span>,<font color="Maroon" title="c12">x</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_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">M</font> &amp; <span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c15">k</font>,<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c13">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E2:116"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="matrix13.html#E4:116"><i><font color="Green" title="E102">A4</font></i></a>, <a class="txt" href="matrix13.html#E5:116"><i><font color="Green" title="E103">A5</font></i></a>, <a class="txt" href="matrix13.html#E5:116_2"><i><font color="Green" title="E105">A14</font></i></a>, <a class="txt" href="matrix13.html#E7:116_2"><i><font color="Green" title="E106">A15</font></i></a>, <a class="txt" href="matrix13.html#E1:116_2_1"><i><font color="Green" title="E107">A16</font></i></a>, <a class="ref" href="matrix13.html#D1" target="_self" title="MATRIX13:def.1">Def1</a>, <a class="ref" href="matrix13.html#T17" target="_self" title="MATRIX13:th.17">Th17</a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>;<br/><a NAME="E4:116_2_1"/>
( <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">x</font> or <span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> )
 ;<br/><b>hence </b><a NAME="E5:116_2_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><a href="matrix13.html#K7" title="MATRIX13:func.7">EqSegm</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">Q</font></span>)</span>,<font color="Maroon" title="c13">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">K</font>
 <b>by </b><i><a class="txt" href="matrix13.html#E1:116"><i><font color="Green" title="E99">A1</font></i></a>, <a class="txt" href="matrix13.html#E3:116_2"><i><font color="Green" title="E104">A13</font></i></a>, <a class="txt" href="matrix13.html#E3:116_2_1"><i><font color="Green" title="E108">A17</font></i></a>, <a class="ref" href="matrix_1.html#D15" title="MATRIX_1:def.15">MATRIX_1:def 15</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:116_2"/>
contradiction
 <b>by </b><i><a class="txt" href="matrix13.html#E2:116"><i><font color="Green" title="E100">A2</font></i></a>, <a class="txt" href="matrix13.html#E3:116"><i><font color="Green" title="E101">A3</font></i></a>, <a class="txt" href="matrix13.html#E5:116_2"><i><font color="Green" title="E105">A14</font></i></a>, <a class="txt" href="matrix13.html#E7:116_2"><i><font color="Green" title="E106">A15</font></i></a>, <a class="ref" href="matrix_9.html#T53" title="MATRIX_9:th.53">MATRIX_9:53</a></i>;<br/>


</div><b>end;</b></div>


</div>
