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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">A</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <font color="Maroon" title="c3">D</font>;<br/><b>let </b><font color="Maroon" title="c5">P</font>, <font color="Maroon" title="c6">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>set </b><font color="Maroon" title="c7">SP</font> =  <a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c5">P</font>;<br/>
<b>set </b><font color="Maroon" title="c8">SQ</font> =  <a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c6">Q</font>;<br/>
<b>set </b><font color="Maroon" title="c9">I</font> =  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c10">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E2:64"/><i><font color="Green" title="E56">A1</font></i>: 
<font color="Maroon" title="c5">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> <font color="Maroon" title="c10">n</font>
 <b>by </b><i><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">TT43</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:64"/><i><font color="Green" title="E57">A2</font></i>: 
<font color="Maroon" title="c6">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> <font color="Maroon" title="c11">m</font>
 <b>by </b><i><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">TT43</a></i>;<br/>
<a NAME="E5:64"/>
(  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c5">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">P</font></span>)</span> &amp;  <a href="finsop_1.html#K5" title="FINSOP_1:func.5">dom</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c6">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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">Q</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E2:64"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="matrix13.html#E4:64"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a></i>;<br/>
<a NAME="E6:64"/><b>then </b><i><font color="Green" title="E58">A3</font></i>: 
( <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">X</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="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">P</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c6">Q</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">Y</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="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">Q</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T167" title="RELAT_1:th.167">RELAT_1:167</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_1_1"/>
(  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">P</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_1"><b>suppose </b></a><a NAME="E1:64_1_1_1"/><i><font color="Green" title="E59">A4</font></i>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:64_1_1_1"/><b>then </b>
<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="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">P</font></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="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">Q</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></i>;<br/><b>hence </b><a NAME="E3:64_1_1_1"/>
<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="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">P</font></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="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">Q</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E1:64_1_1_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_1_1_2"><b>suppose </b></a><a NAME="E1:64_1_1_2"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">P</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:64_1_1_2"/>
<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="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">P</font></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="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">Q</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:64"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c1">X</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c6">Q</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c2">Y</font></span>)</span></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> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix13.html#E6:64"><i><font color="Green" title="E58">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>


</div>
