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

<b>let </b><font color="Maroon">S</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be  <a href="partfun1.html#V1">total</a>  <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">S</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">BASSIGN</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font></span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="modelc_1.html#NM2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" target="_self">CTLModel</a> <font color="Maroon">R</font>,<font color="Maroon">BASSIGN</font></span>)</span>, <font color="Maroon">g</font> be   <a href="modelc_1.html#NM2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" target="_self">CTLModel</a> <font color="Maroon">R</font>,<font color="Maroon">BASSIGN</font></span>)</span>;<br/>







<b>set </b><font color="Maroon">G</font> =  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font>;<br/>
<b>set </b><font color="Maroon">Q</font> =  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>;<br/>
<i><font color="Green">E27</font></i>: <span class="p1">(<span class="default"><a href="modelc_1.html#K60" target="_self">TransEG</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> = 
 <a href="modelc_1.html#K56" target="_self">SigFaxTau</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">R</font>,<font color="Maroon">BASSIGN</font>
<b>by </b><i><a class="txt" href="modelc_1.html#E37"><i><font color="Green">Lemma127</font></i></a></i>
<br/>.= 
 <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E2:169"/><i><font color="Green">E28</font></i>: 
( ( for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> iff <font color="Olive">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> ) ) implies  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font> <a href="abian.html#R2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K60" target="_self">TransEG</a> <font color="Maroon">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="169_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:169_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> iff <font color="Olive">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> )
 ;<br/>

<a NAME="E2:169_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="169_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:169_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font>
 ;<br/>

<b>consider </b><font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:169_2_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:169_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#D23" target="_self">Def23</a></i>;<br/>
<a NAME="E5:169_2_1"/>
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E45"><i><font color="Green">Lemma135</font></i></a>, <a class="ref" href="modelc_1.html#D21" target="_self">Def21</a></i>;<br/>
<b>hence </b><a NAME="E6:169_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E44"><i><font color="Green">Lemma134</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:169_2"/>
for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> holds <br/><font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="169_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:169_2_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:169_2_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:169_2_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#D23" target="_self">Def23</a></i>;<br/>
<a NAME="E5:169_2_2"/>
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E45"><i><font color="Green">Lemma135</font></i></a>, <a class="ref" href="modelc_1.html#D21" target="_self">Def21</a></i>;<br/>
<b>hence </b><a NAME="E6:169_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E44"><i><font color="Green">Lemma134</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:169_2"/><b>then </b>
 <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K60" target="_self">TransEG</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E39"><i><font color="Green">Lemma129</font></i></a>, <a class="ref" href="modelc_1.html#D22" target="_self">Def22</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E5:169_2"/>
 <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font> <a href="abian.html#R2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K60" target="_self">TransEG</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="abian.html#D4">ABIAN:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:169"/>
(  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font> <a href="abian.html#R2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K60" target="_self">TransEG</a> <font color="Maroon">f</font> implies for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> iff <font color="Olive">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="169_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:169_3"/>
 <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font> <a href="abian.html#R2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K60" target="_self">TransEG</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:169_3"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E39"><i><font color="Green">Lemma129</font></i></a>, <a class="ref" href="abian.html#D4">ABIAN:def 4</a></i>;<br/>
<a NAME="E3:169_3"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> iff <font color="Olive">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="169_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/>

<b>thus </b><a NAME="E1:169_3_1"/>
( <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> implies <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="169_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:169_3_1_1"/>
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:169_3_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="modelc_1.html#D21" target="_self">Def21</a></i>;<br/>
<b>then consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:169_3_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>and </b><br/><a NAME="E5:169_3_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">t</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
 ;<br/>
<b>thus </b><a NAME="E6:169_3_1_1"/>
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#D22" target="_self">Def22</a>, <a class="ref" href="modelc_1.html#D23" target="_self">Def23</a></i>;<br/>


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

<b>assume </b><a NAME="E2:169_3_1"/>
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
 ;<br/>

<a NAME="E3:169_3_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="modelc_1.html#D21" target="_self">Def21</a></i>;<br/>
<b>then consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E5:169_3_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> )
 ;<br/>
<b>thus </b><a NAME="E6:169_3_1"/>
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#D22" target="_self">Def22</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:169_3"/>
for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> iff <font color="Olive">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:169"/>
( ( for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> iff <font color="Olive">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K55" target="_self">Fax</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> ) ) iff  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font> <a href="abian.html#R2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K60" target="_self">TransEG</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="txt" href="modelc_1.html#E40"><i><font color="Green">Lemma130</font></i></a></i>;<br/>


</div>
