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

<b>let </b><font color="Maroon" title="c1">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="c2">A</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <font color="Maroon" title="c1">D</font>;<br/><b>let </b><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">Q</font>, <font color="Maroon" title="c5">P1</font>, <font color="Maroon" title="c6">Q1</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="E1:66"/><i><font color="Green" title="E58">A1</font></i>: 
( <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  iff <font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>and </b><br/><a NAME="E2:66"/><i><font color="Green" title="E59">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="c3">P</font>,<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">Q1</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon" title="c7">S</font> =  <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">Q1</font>;<br/>
<div><i><font color="Green" title="E60">SS</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:66_1_1"/>
( <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_1"><b>suppose </b></a><a NAME="E1:66_1_1_1"/>
<font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:66_1_1_1"/>
( <font color="Maroon" title="c3">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="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">P1</font></span>)</span> &amp; <font color="Maroon" title="c4">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="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">Q1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:66"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="66_1_1_2"><b>suppose </b></a><a NAME="E1:66_1_1_2"/><i><font color="Green" title="E61">A3</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:66_1_1_2"/><b>then </b>
( <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <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="c2">A</font>,<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">Q1</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#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">Q1</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#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">Q1</font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:66"><i><font color="Green" title="E58">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:66_1_1_2"/><b>then </b><i><font color="Green" title="E62">A4</font></i>: 
( <font color="Maroon" title="c3">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> <span class="p2">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">Q1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c4">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> <span class="p2">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">Q1</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E2:66"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T138" title="ZFMISC_1:th.138">ZFMISC_1:138</a></i>;<br/><a NAME="E4:66_1_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">Q1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">Q1</font></span>)</span> <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">P1</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E1:66_1_1_2"><i><font color="Green" title="E61">A3</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a></i>;<br/><b>hence </b><a NAME="E5:66_1_1_2"/>
( <font color="Maroon" title="c3">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="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">P1</font></span>)</span> &amp; <font color="Maroon" title="c4">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="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c6">Q1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix13.html#E3:66_1_1_2"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="matrix13.html#T1" target="_self" title="MATRIX13:th.1">TT16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">k</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="E5:66"/><i><font color="Green" title="E61">A3</font></i>: 
<font color="Maroon" title="c5">P1</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="c8">k</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="c9">k</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="E7:66"/><i><font color="Green" title="E62">A4</font></i>: 
<font color="Maroon" title="c6">Q1</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="c9">k</font>
 <b>by </b><i><a class="ref" href="matrix13.html#T43" target="_self" title="MATRIX13:th.43">TT43</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">SP</font> =  <a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c5">P1</font>;<br/>
<b>set </b><font color="Maroon" title="c11">SQ</font> =  <a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c6">Q1</font>;<br/>
<a NAME="E8:66"/><i><font color="Green" title="E63">A5</font></i>: 
(  <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">P1</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">P1</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#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c5">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P1</font> &amp;  <a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c5">P1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">without_repeated_line</a> &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">Q1</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">Q1</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#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c6">Q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Q1</font> &amp;  <a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c6">Q1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">without_repeated_line</a> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E5:66"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="matrix13.html#E7:66"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="finseq_3.html#T45" title="FINSEQ_3:th.45">FINSEQ_3:45</a>, <a class="ref" href="finseq_3.html#T99" title="FINSEQ_3:th.99">FINSEQ_3:99</a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/>
<a NAME="E9:66"/><b>then </b><i><font color="Green" title="E64">A6</font></i>: 
( <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c5">P1</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P1</font> &amp; not 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P1</font> &amp; <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c6">Q1</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Q1</font> &amp; not 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q1</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T144" title="RELAT_1:th.144">RELAT_1:144</a>, <a class="ref" href="pscomp_1.html#D1" title="PSCOMP_1:def.1">PSCOMP_1:def 1</a></i>;<br/>
<a NAME="E10:66"/><b>then </b>
( <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c5">P1</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">P</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c5">P1</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; not 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c5">P1</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">P</font> &amp; <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c6">Q1</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">Q</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c6">Q1</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; not 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c6">Q1</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">Q</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">P2</font> = <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c5">P1</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c13">Q2</font> = <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c6">Q1</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">Q</font> as  <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>  <b>by </b><i><a class="ref" href="pscomp_1.html#D1" title="PSCOMP_1:def.1">PSCOMP_1:def 1</a></i>;<br/>
<a NAME="E12:66"/>
( <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c5">P1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c12">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> &amp; <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c6">Q1</font></span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c13">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q</font> &amp; <font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c12">P2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a>  &amp; <font color="Maroon" title="c4">Q</font>,<font color="Maroon" title="c13">Q2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a>  )
 
<b>by </b><i><a class="txt" href="matrix13.html#E8:66"><i><font color="Green" title="E63">A5</font></i></a>, <a class="txt" href="matrix13.html#E3:66"><i><font color="Green" title="E60">SS</font></i></a>, <a class="ref" href="funct_1.html#T164" title="FUNCT_1:th.164">FUNCT_1:164</a>, <a class="ref" href="card_1.html#T60" title="CARD_1:th.60">CARD_1:60</a></i>;<br/>
<a NAME="E13:66"/><b>then </b>
(  <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">Q1</font></span>)</span>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c12">P2</font>,<font color="Maroon" title="c13">Q2</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">P</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="c12">P2</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">Q</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="c13">Q2</font> )
 
<b>by </b><i><a class="txt" href="matrix13.html#E9:66"><i><font color="Green" title="E64">A6</font></i></a>, <a class="ref" href="matrix13.html#T56" target="_self" title="MATRIX13:th.56">TT56</a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/>
<b>hence </b><a NAME="E14:66"/>
 ex <font color="Olive" title="b1">P2</font>, <font color="Olive" title="b2">Q2</font> being <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>  st <br/>( <font color="Olive" title="b1">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P1</font> &amp; <font color="Olive" title="b2">Q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Q1</font> &amp; <font color="Olive" title="b1">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c5">P1</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b2">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix13.html#K3" title="MATRIX13:func.3">Sgm</a> <font color="Maroon" title="c6">Q1</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">Q</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b1">P2</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="c3">P</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">Q2</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="c4">Q</font> &amp;  <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <span class="p1">(<span class="default"><a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">Q1</font></span>)</span>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K4" title="MATRIX13:func.4">Segm</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">P2</font>,<font color="Olive" title="b2">Q2</font> )
 <b>by </b><i><a class="txt" href="matrix13.html#E9:66"><i><font color="Green" title="E64">A6</font></i></a></i>;<br/>


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