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<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">h</font> = <font color="Maroon">f</font> <a href="modelc_1.html#K29" target="_self">EU</a> <font color="Maroon">g</font>;<br/>
<a NAME="E1:179"/>
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">f</font> <a href="modelc_1.html#K29" target="_self">EU</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#K61" target="_self">Foax</a> <font color="Maroon">g</font>,<font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K29" target="_self">EU</a> <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/>
<a NAME="E2:179"/><b>then </b>
 <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K29" target="_self">EU</a> <font color="Maroon">g</font></span>)</span> <a href="abian.html#R2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K63" target="_self">TransEU</a> <font color="Maroon">f</font>,<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/>
<a NAME="E3:179"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="knaster.html#K1">lfp</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K63" target="_self">TransEU</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K29" target="_self">EU</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="knaster.html#T10">KNASTER:10</a></i>;<br/>
<b>set </b><font color="Maroon">p</font> =  <a href="modelc_1.html#K54" target="_self">Tau</a> <span class="p1">(<span class="default"><a href="knaster.html#K1">lfp</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="modelc_1.html#K63" target="_self">TransEU</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">R</font>,<font color="Maroon">BASSIGN</font>;<br/>
<a NAME="E4:179"/><i><font color="Green">E28</font></i>: 
 <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" target="_self">Tau</a> <span class="p2">(<span class="default"><a href="knaster.html#K1">lfp</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><a href="modelc_1.html#K63" target="_self">TransEU</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">R</font>,<font color="Maroon">BASSIGN</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="knaster.html#K1">lfp</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K63" target="_self">TransEU</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:179"/>
 <a href="knaster.html#K1">lfp</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K63" target="_self">TransEU</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="abian.html#R2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K63" target="_self">TransEU</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="knaster.html#T6">KNASTER:6</a></i>;<br/>
<a NAME="E6:179"/><b>then </b><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>  <a href="modelc_1.html#K54" target="_self">Tau</a> <span class="p1">(<span class="default"><a href="knaster.html#K1">lfp</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="modelc_1.html#K63" target="_self">TransEU</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">R</font>,<font color="Maroon">BASSIGN</font> iff <font color="Olive">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K61" target="_self">Foax</a> <font color="Maroon">g</font>,<font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K54" target="_self">Tau</a> <span class="p2">(<span class="default"><a href="knaster.html#K1">lfp</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><a href="modelc_1.html#K63" target="_self">TransEU</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">R</font>,<font color="Maroon">BASSIGN</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="modelc_1.html#E46"><i><font color="Green">Lemma136</font></i></a>, <a class="txt" href="modelc_1.html#E44"><i><font color="Green">Lemma134</font></i></a></i>;<br/>
<a NAME="E7:179"/>
 <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K29" target="_self">EU</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" target="_self">Tau</a> <span class="p2">(<span class="default"><a href="knaster.html#K1">lfp</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><a href="modelc_1.html#K63" target="_self">TransEU</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">R</font>,<font color="Maroon">BASSIGN</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="179_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:179_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"><font color="Maroon">f</font> <a href="modelc_1.html#K29" target="_self">EU</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<b>then 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="E3:179_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:179_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> <a href="modelc_1.html#K29" target="_self">EU</a> <font color="Maroon">g</font>
 ;<br/>
<a NAME="E5:179_1"/>
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K54" target="_self">Tau</a> <span class="p1">(<span class="default"><a href="knaster.html#K1">lfp</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="modelc_1.html#K63" target="_self">TransEU</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">R</font>,<font color="Maroon">BASSIGN</font>
 
<b>by </b><i>, <a class="txt" href="modelc_1.html#E47"><i><font color="Green">Lemma137</font></i></a>, <a class="ref" href="modelc_1.html#D22" target="_self">Def22</a></i>;<br/>
<b>hence </b><a NAME="E6:179_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#K54" target="_self">Tau</a> <span class="p2">(<span class="default"><a href="knaster.html#K1">lfp</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><a href="modelc_1.html#K63" target="_self">TransEU</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span></span>)</span>,<font color="Maroon">R</font>,<font color="Maroon">BASSIGN</font></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E48"><i><font color="Green">Lemma138</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:179"/>
 <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K29" target="_self">EU</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="knaster.html#K1">lfp</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K63" target="_self">TransEU</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E45"><i><font color="Green">Lemma135</font></i></a>, <a class="txt" href="modelc_1.html#E46"><i><font color="Green">Lemma136</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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