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

<b>let </b><font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c2">C2</font> be   <a href="coh_sp.html#NM1" title="COH_SP:NM.1">Coherence_Space</a>;<br/><b>let </b><font color="Maroon" title="c3">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">C1</font>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:84"/><i><font color="Green" title="E44">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Olive" title="b1">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>and </b><br/><a NAME="E2:84"/><i><font color="Green" title="E45">A2</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font>  st <font color="Olive" title="b1">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> holds <br/>for <font color="Olive" title="b3">y1</font>, <font color="Olive" title="b4">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b3">y1</font>,<font color="Olive" title="b4">y2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font>
 <b>and </b><br/><a NAME="E3:84"/><i><font color="Green" title="E46">A3</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font>  st <font color="Olive" title="b1">a</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> holds <br/>for <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c3">X</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:84"/><i><font color="Green" title="E47">A4</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">C1</font> &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E6:84"/><i><font color="Green" title="E48">A5</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:84_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:84_1"/><i><font color="Green" title="E49">A6</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> )
 <b>by </b><i><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="c7">a</font> = <font color="Maroon" title="c6">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="cohsp_1.html#E3:84_1"><i><font color="Green" title="E49">A6</font></i></a></i>;<br/>
<a NAME="E5:84_1"/><i><font color="Green" title="E50">A7</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c7">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="cohsp_1.html#E3:84_1"><i><font color="Green" title="E49">A6</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">z</font>, <font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:84_1_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">x</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c10">z1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:84_1_1"/><i><font color="Green" title="E51">A8</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">z1</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c10">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c7">a</font> )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></i>;<br/><b>assume </b><a NAME="E4:84_1_1"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">x</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c11">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:84_1_1"/><i><font color="Green" title="E52">A9</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c9">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c11">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c7">a</font> )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></i>;<br/><a NAME="E7:84_1_1"/>
( <font color="Maroon" title="c10">z1</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> &amp; <font color="Maroon" title="c10">z1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c11">y1</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E3:84_1_1"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="cohsp_1.html#E6:84_1_1"><i><font color="Green" title="E52">A9</font></i></a>, <a class="ref" href="finsub_1.html#T10" title="FINSUB_1:th.10">FINSUB_1:10</a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:84_1_1"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c9">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E2:84"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="cohsp_1.html#E3:84_1_1"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="cohsp_1.html#E6:84_1_1"><i><font color="Green" title="E52">A9</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:84_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C2</font>
 <b>by </b><i><a class="ref" href="coh_sp.html#T6" title="COH_SP:th.6">COH_SP:6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:84"/><i><font color="Green" title="E49">A10</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="cohsp_1.html#V6" title="COHSP_1:attr.6">c=-monotone</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="cohsp_1.html#D12" target="_self" title="COHSP_1:def.12">COHSP_1:def 12</a><br/>

<b>assume </b><a NAME="E1:84_2"/><i><font color="Green" title="E50">A11</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c5">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">b</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c6">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a></i>;<br/>
<a NAME="E3:84_2"/>
 <a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c7">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c8">b</font>
 
<b>by </b><i><a class="txt" href="cohsp_1.html#E1:84_2"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="finsub_1.html#T23" title="FINSUB_1:th.23">FINSUB_1:23</a></i>;<br/>
<a NAME="E4:84_2"/><b>then </b>
( <font color="Maroon" title="c3">X</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c7">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c8">b</font></span>)</span> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c7">a</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<b>hence </b><a NAME="E5:84_2"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:84_3"/><i><font color="Green" title="E50">A12</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> )
 ;<br/><a NAME="E2:84_3"/><b>then </b>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c5">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:84_3"/><i><font color="Green" title="E51">A13</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">a'</font> = <font color="Maroon" title="c5">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="cohsp_1.html#E1:84_3"><i><font color="Green" title="E50">A12</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c7">x</font>;<br/><a NAME="E6:84_3"/>
<font color="Maroon" title="c9">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E4:84_3"><i><font color="Green" title="E51">A13</font></i></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/><a NAME="E7:84_3"/><b>then </b>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> &amp; <font color="Maroon" title="c9">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="cohsp_1.html#E1:84_3"><i><font color="Green" title="E50">A12</font></i></a>, <a class="txt" href="cohsp_1.html#E4:84_3"><i><font color="Green" title="E51">A13</font></i></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/><a NAME="E8:84_3"/><b>then </b>
( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c9">x</font></span>)</span> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c9">x</font> )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/><b>hence </b><a NAME="E9:84_3"/>
( <font color="Maroon" title="c9">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; <font color="Maroon" title="c9">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E4:84_3"><i><font color="Green" title="E51">A13</font></i></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a>, <a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></i>;<br/><b>let </b><font color="Maroon" title="c10">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E10:84_3"/><i><font color="Green" title="E52">A14</font></i>: 
( <font color="Maroon" title="c10">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">c</font> )
 ;<br/><a NAME="E11:84_3"/><b>then </b>
<font color="Maroon" title="c10">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a'</font>
 ;<br/><a NAME="E12:84_3"/><b>then </b>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="classes1.html#D1" title="CLASSES1:def.1">CLASSES1:def 1</a></i>;<br/><a NAME="E13:84_3"/><b>then </b>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c10">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="cohsp_1.html#E10:84_3"><i><font color="Green" title="E52">A14</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:84_3"/><i><font color="Green" title="E53">A15</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c10">c</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></i>;<br/><a NAME="E16:84_3"/>
 <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c10">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E10:84_3"><i><font color="Green" title="E52">A14</font></i></a>, <a class="ref" href="finsub_1.html#T23" title="FINSUB_1:th.23">FINSUB_1:23</a></i>;<br/><a NAME="E17:84_3"/><b>then </b>
( <font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c8">a'</font> )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E4:84_3"><i><font color="Green" title="E51">A13</font></i></a>, <a class="txt" href="cohsp_1.html#E15:84_3"><i><font color="Green" title="E53">A15</font></i></a></i>;<br/><a NAME="E18:84_3"/><b>then </b>
( <font color="Maroon" title="c9">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C1</font> )
 <b>by </b><i><a class="ref" href="finsub_1.html#T10" title="FINSUB_1:th.10">FINSUB_1:10</a></i>;<br/><a NAME="E19:84_3"/><b>then </b>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z</font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E3:84"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="cohsp_1.html#E4:84_3"><i><font color="Green" title="E51">A13</font></i></a>, <a class="txt" href="cohsp_1.html#E15:84_3"><i><font color="Green" title="E53">A15</font></i></a></i>;<br/><b>hence </b><a NAME="E20:84_3"/>
<font color="Maroon" title="c9">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">c</font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E15:84_3"><i><font color="Green" title="E53">A15</font></i></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">f</font> as  <a href="cohsp_1.html#V9" title="COHSP_1:attr.9">U-stable</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="cohsp_1.html#E6:84"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="cohsp_1.html#E7:84"><i><font color="Green" title="E49">A10</font></i></a>, <a class="ref" href="cohsp_1.html#T23" target="_self" title="COHSP_1:th.23">Th23</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">f</font>
;<br/>

<b>thus </b><a NAME="E10:84"/>
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cohsp_1.html#K8" title="COHSP_1:func.8">Trace</a> <font color="Maroon" title="c5">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D2" title="RELAT_1:def.2">RELAT_1:def 2</a><br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:84_4_1"/><i><font color="Green" title="E50">A16</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ;<br/><a NAME="E2:84_4_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">C1</font>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c2">C2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">a'</font> = <font color="Maroon" title="c6">a</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="txt" href="cohsp_1.html#E1:84"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="cohsp_1.html#E1:84_4_1"><i><font color="Green" title="E50">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="E4:84_4_1"/>
<font color="Maroon" title="c8">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/><a NAME="E5:84_4_1"/><b>then </b><i><font color="Green" title="E51">A17</font></i>: 
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c6">a</font></span>)</span> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c6">a</font></span>)</span> )
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="cohsp_1.html#E1:84_4_1"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:84_4_1_1"/><i><font color="Green" title="E52">A18</font></i>: 
( <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c9">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a'</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">c</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">c'</font> = <font color="Maroon" title="c9">c</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a></i>;<br/><a NAME="E3:84_4_1_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c10">c'</font></span>)</span>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="cohsp_1.html#E1:84_4_1_1"><i><font color="Green" title="E52">A18</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:84_4_1_1"/><i><font color="Green" title="E53">A19</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c10">c'</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></i>;<br/><a NAME="E6:84_4_1_1"/><i><font color="Green" title="E54">A20</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">c</font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84_4_1_1"><i><font color="Green" title="E53">A19</font></i></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/><a NAME="E7:84_4_1_1"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a'</font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E1:84_4_1_1"><i><font color="Green" title="E52">A18</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><a NAME="E8:84_4_1_1"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E3:84"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="cohsp_1.html#E1:84_4_1"><i><font color="Green" title="E50">A16</font></i></a>, <a class="txt" href="cohsp_1.html#E5:84_4_1_1"><i><font color="Green" title="E53">A19</font></i></a></i>;<br/><b>hence </b><a NAME="E9:84_4_1_1"/>
<font color="Maroon" title="c8">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">c</font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E1:84_4_1_1"><i><font color="Green" title="E52">A18</font></i></a>, <a class="txt" href="cohsp_1.html#E6:84_4_1_1"><i><font color="Green" title="E54">A20</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:84_4_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</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="cohsp_1.html#K8" title="COHSP_1:func.8">Trace</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="cohsp_1.html#E5:84_4_1"><i><font color="Green" title="E51">A17</font></i></a>, <a class="ref" href="cohsp_1.html#T32" target="_self" title="COHSP_1:th.32">Th32</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:84_4"/><i><font color="Green" title="E50">A21</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</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="cohsp_1.html#K8" title="COHSP_1:func.8">Trace</a> <font color="Maroon" title="c5">f</font>
 ;<br/>

<a NAME="E3:84_4"/><b>then </b>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> )
 
<b>by </b><i><a class="ref" href="cohsp_1.html#T32" target="_self" title="COHSP_1:th.32">Th32</a></i>;<br/>
<a NAME="E4:84_4"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c6">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:84_4"/><i><font color="Green" title="E51">A22</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c6">a</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c6">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="cohsp_1.html#E2:84_4"><i><font color="Green" title="E50">A21</font></i></a>, <a class="ref" href="cohsp_1.html#T32" target="_self" title="COHSP_1:th.32">Th32</a></i>;<br/>
<a NAME="E8:84_4"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c9">a</font>
 
<b>by </b><i><a class="txt" href="cohsp_1.html#E6:84_4"><i><font color="Green" title="E51">A22</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c8">x</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> <b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/>
<a NAME="E10:84_4"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/>
<a NAME="E11:84_4"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Maroon" title="c10">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="cohsp_1.html#E6:84_4"><i><font color="Green" title="E51">A22</font></i></a>, <a class="ref" href="relat_1.html#D13" title="RELAT_1:def.13">RELAT_1:def 13</a></i>;<br/>
<a NAME="E12:84_4"/><b>then </b>
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c10">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">a</font> )
 
<b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="cohsp_1.html#E6:84_4"><i><font color="Green" title="E51">A22</font></i></a>, <a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E13:84_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="cohsp_1.html#E2:84_4"><i><font color="Green" title="E50">A21</font></i></a>, <a class="txt" href="cohsp_1.html#E6:84_4"><i><font color="Green" title="E51">A22</font></i></a>, <a class="ref" href="cohsp_1.html#T32" target="_self" title="COHSP_1:th.32">Th32</a></i>;<br/>


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

<b>thus </b><a NAME="E11:84"/>
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C1</font> holds  <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="cohsp_1.html#K3" title="COHSP_1:func.3">Fin</a> <font color="Olive" title="b1">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="cohsp_1.html#E5:84"><i><font color="Green" title="E47">A4</font></i></a></i>;<br/>


</div>
