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

<b>let </b><font color="Maroon" title="c1">S</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">R</font> be  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>;<br/><b>let </b><font color="Maroon" title="c3">BASSIGN</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="modelc_1.html#NM2" title="MODELC_1:NM.2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" title="MODELC_1:func.50">CTLModel</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span>;<br/>







<b>set </b><font color="Maroon" title="c5">T</font> =  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font>;<br/>
<b>set </b><font color="Maroon" title="c6">g</font> =  <a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font>;<br/>
<a NAME="E1:152"/><i><font color="Green" title="E114">A1</font></i>: 
 <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s</font> where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> : <span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  </span>}</span> 
 
<b>by </b><i><a class="txt" href="modelc_1.html#E137"><i><font color="Green" title="E109">Lm46</font></i></a></i>;<br/>
<a NAME="E2:152"/>
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> holds <br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <span class="p3">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="152_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:152_1"/><i><font color="Green" title="E115">A2</font></i>: 
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c7">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="modelc_1.html#E1:152_1"><i><font color="Green" title="E115">A2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="152_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:152_1_1"/>
( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font> or  not <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="152_1_1_1"><b>suppose </b></a><a NAME="E1:152_1_1_1"/><i><font color="Green" title="E116">A3</font></i>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E3:152_1_1_1"/><i><font color="Green" title="E117">A4</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font>
 <b>and </b><br/><a NAME="E4:152_1_1_1"/><i><font color="Green" title="E118">A5</font></i>: 
<span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:152"><i><font color="Green" title="E114">A1</font></i></a></i>;<br/><span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <span class="p3">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font> = 
<span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font>
<b>by </b><i><a class="ref" href="modelc_1.html#D64" target="_self" title="MODELC_1:def.64">Def64</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="modelc_1.html#E1:152_1_1_1"><i><font color="Green" title="E116">A3</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>
;<br/><b>hence </b><a NAME="E6:152_1_1_1"/>
<span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <span class="p3">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:152_1_1_1"><i><font color="Green" title="E117">A4</font></i></a>, <a class="txt" href="modelc_1.html#E4:152_1_1_1"><i><font color="Green" title="E118">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="152_1_1_2"><b>suppose </b></a><a NAME="E1:152_1_1_2"/><i><font color="Green" title="E116">A6</font></i>: 
not <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font>
 ;<br/><div class="add"><a NAME="E2:152_1_1_2"/><i><font color="Green" title="E117">A7</font></i>: 
<span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="152_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:152_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 ;<br/>

<a NAME="E2:152_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:152_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:152"><i><font color="Green" title="E114">A1</font></i></a>, <a class="txt" href="modelc_1.html#E1:152_1_1_2"><i><font color="Green" title="E116">A6</font></i></a></i>;<br/>


</div><b>end;</b></div><span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <span class="p3">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font> = 
<span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font>
<b>by </b><i><a class="ref" href="modelc_1.html#D64" target="_self" title="MODELC_1:def.64">Def64</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:152_1_1_2"><i><font color="Green" title="E116">A6</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>
;<br/><b>hence </b><a NAME="E4:152_1_1_2"/>
<span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <span class="p3">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span>,<font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E2:152_1_1_2"><i><font color="Green" title="E117">A7</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:152"/><b>then </b>
 <a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">S</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="modelc_1.html#K33" title="MODELC_1:func.33">Fid</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span>,<font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/>
<b>hence </b><a NAME="E4:152"/>
 <a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E139"><i><font color="Green" title="E111">Lm48</font></i></a></i>;<br/>


</div>
