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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="funct_1.html#V2">one-to-one</a> <a href="funct_1.html#NM1">Function</a>;<br/>

<a NAME="E1:8"/><i><font color="Green">E9</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="finseq_2.html#K1">idseq</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<b>assume </b><a NAME="E2:8"/><i><font color="Green">E10</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> 2 &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> 2 )
 ;<br/>

<a NAME="E3:8"/><i><font color="Green">E11</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E2:8"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E4:8"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E5:8"/><b>then </b><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="txt" href="matrix_9.html#E2:8"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E6:8"/><i><font color="Green">E13</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E2:8"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E7:8"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E8:8"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="matrix_9.html#E2:8"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>
<a NAME="E9:8"/><b>then </b><i><font color="Green">E14</font></i>: 
( ( <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> 2 ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> 1 ) )
 
<b>by </b><i><a class="txt" href="matrix_9.html#E5:8"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>, <a class="txt" href="matrix_9.html#E3:8"><i><font color="Green">E11</font></i></a>, <a class="txt" href="matrix_9.html#E6:8"><i><font color="Green">E13</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> 2 ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="matrix_9.html#E9:8"><i><font color="Green">E14</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1"><b>suppose </b></a><a NAME="E1:8_1_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:8_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> 2 holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:8_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> 2
 ;<br/>

<a NAME="E2:8_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>
<b>hence </b><a NAME="E3:8_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="matrix_9.html#E1:8_1_1"><i><font color="Green">E15</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:8_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 2</span>)</span> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K3">Rev</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 2</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a>, <a class="txt" href="matrix_9.html#E2:8"><i><font color="Green">E10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2"><b>suppose </b></a><a NAME="E1:8_1_2"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 2 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> 1 )
 ;<br/><div class="add"><a NAME="E2:8_1_2"/><i><font color="Green">E16</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_9.html#E2:8"><i><font color="Green">E10</font></i></a>, <a class="txt" href="matrix_9.html#E1:8"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finseq_5.html#T60">FINSEQ_5:60</a></i>;<br/><a NAME="E3:8_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> 2 holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 2</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:8_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> 2
 ;<br/>

<a NAME="E2:8_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>
<b>hence </b><a NAME="E3:8_1_2_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K3">Rev</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 2</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="matrix_9.html#E1:8_1_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="matrix_9.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:8_1_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 2</span>)</span> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_5.html#K3">Rev</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> 2</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix_9.html#E2:8_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a>, <a class="txt" href="matrix_9.html#E2:8"><i><font color="Green">E10</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
