<?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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">R</font> being <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/> for <font color="Olive" title="b2">BASSIGN</font> being 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/> for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">g</font>, <font color="Olive" title="b5">h</font> being  <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="Olive" title="b1">R</font>,<font color="Olive" title="b2">BASSIGN</font></span>)</span> holds <br/> ( ( for <font color="Olive" title="b6">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b5">h</font> iff <font color="Olive" title="b6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Olive" title="b4">g</font>,<font color="Olive" title="b3">f</font>,<font color="Olive" title="b5">h</font> ) ) iff  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Olive" title="b5">h</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K63" title="MODELC_1:func.63">TransEU</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">g</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">BASSIGN</font> being 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/> for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font>, <font color="Olive" title="b4">h</font> being  <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="Olive" title="b1">BASSIGN</font></span>)</span> holds <br/> ( ( for <font color="Olive" title="b5">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b5">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b4">h</font> iff <font color="Olive" title="b5">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Olive" title="b3">g</font>,<font color="Olive" title="b2">f</font>,<font color="Olive" title="b4">h</font> ) ) iff  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Olive" title="b4">h</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K63" title="MODELC_1:func.63">TransEU</a> <font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">g</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being  <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> holds <br/> ( ( for <font color="Olive" title="b4">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b4">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Olive" title="b3">h</font> iff <font color="Olive" title="b4">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">h</font> ) ) iff  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Olive" title="b3">h</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K63" title="MODELC_1:func.63">TransEU</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> )</span><br/><b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font>, <font color="Maroon" title="c6">h</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font> iff <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font> ) ) iff  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c6">h</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K63" title="MODELC_1:func.63">TransEU</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> )</span><br/>







<b>set </b><font color="Maroon" title="c7">H</font> =  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c6">h</font>;<br/>
<b>set </b><font color="Maroon" title="c8">Q</font> =  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font></span>)</span>;<br/>
<i><font color="Green" title="E135">A1</font></i>: <span class="p1">(<span class="default"><a href="modelc_1.html#K63" title="MODELC_1:func.63">TransEU</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c6">h</font></span>)</span> = 
 <a href="modelc_1.html#K62" title="MODELC_1:func.62">SigFoaxTau</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c6">h</font></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font>
<b>by </b><i><a class="ref" href="modelc_1.html#D73" target="_self" title="MODELC_1:def.73">Def73</a></i>
<br/>.= 
 <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font></span>)</span>
<b>by </b><i><a class="ref" href="modelc_1.html#T31" target="_self" title="MODELC_1:th.31">Th31</a></i>
;<br/>
<a NAME="E2:178"/><i><font color="Green" title="E136">A2</font></i>: 
( ( for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font> iff <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font> ) ) implies  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c6">h</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K63" title="MODELC_1:func.63">TransEU</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:178_2"/><i><font color="Green" title="E137">A3</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font> iff <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c6">h</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K63" title="MODELC_1:func.63">TransEU</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span><br/>

<a NAME="E2:178_2"/><i><font color="Green" title="E138">A4</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>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c6">h</font> holds <br/><font color="Olive" title="b1">x</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> <span class="p1">(<span class="default"><a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">x</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="c6">h</font> implies <font color="Maroon" title="c9">x</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> <span class="p1">(<span class="default"><a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:178_2_1"/><i><font color="Green" title="E139">A5</font></i>: 
<font color="Maroon" title="c9">x</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="c6">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">x</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> <span class="p1">(<span class="default"><a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font></span>)</span></span><br/>

<b>consider </b><font color="Maroon" title="c10">s</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="E3:178_2_1"/><i><font color="Green" title="E140">A6</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font>
 <b>and </b><br/><a NAME="E4:178_2_1"/><i><font color="Green" title="E141">A7</font></i>: 
<font color="Maroon" title="c10">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font>
 <b>by </b><i><a class="txt" href="#E1:178_2_1"><i><font color="Green" title="E139">A5</font></i></a></i>;<br/>
<a NAME="E5:178_2_1"/>
<font color="Maroon" title="c10">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>
 
<b>by </b><i><a class="txt" href="#E1:178_2"><i><font color="Green" title="E137">A3</font></i></a>, <a class="txt" href="#E4:178_2_1"><i><font color="Green" title="E141">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:178_2_1"/>
<font color="Maroon" title="c9">x</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> <span class="p1">(<span class="default"><a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:178_2_1"><i><font color="Green" title="E140">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:178_2"/>
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>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font></span>)</span> holds <br/><font color="Olive" title="b1">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="c6">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">x</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> <span class="p1">(<span class="default"><a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font></span>)</span> implies <font color="Maroon" title="c9">x</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="c6">h</font> )</span><br/>

<b>assume </b><a NAME="E1:178_2_2"/><i><font color="Green" title="E139">A8</font></i>: 
<font color="Maroon" title="c9">x</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> <span class="p1">(<span class="default"><a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">x</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="c6">h</font></span><br/>

<b>consider </b><font color="Maroon" title="c10">s</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="E3:178_2_2"/><i><font color="Green" title="E140">A9</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font>
 <b>and </b><br/><a NAME="E4:178_2_2"/><i><font color="Green" title="E141">A10</font></i>: 
<font color="Maroon" title="c10">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>
 <b>by </b><i><a class="txt" href="#E1:178_2_2"><i><font color="Green" title="E139">A8</font></i></a></i>;<br/>
<a NAME="E5:178_2_2"/>
<font color="Maroon" title="c10">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font>
 
<b>by </b><i><a class="txt" href="#E1:178_2"><i><font color="Green" title="E137">A3</font></i></a>, <a class="txt" href="#E4:178_2_2"><i><font color="Green" title="E141">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:178_2_2"/>
<font color="Maroon" title="c9">x</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="c6">h</font>
 <b>by </b><i><a class="txt" href="#E3:178_2_2"><i><font color="Green" title="E140">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:178_2"/><b>then </b>
 <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:178_2"><i><font color="Green" title="E138">A4</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E5:178_2"/>
 <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c6">h</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K63" title="MODELC_1:func.63">TransEU</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>
 <b>by </b><i><a class="txt" href="#E1:178"><i><font color="Green" title="E135">A1</font></i></a>, <a class="ref" href="abian.html#D4" title="ABIAN:def.4">ABIAN:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:178"/>
(  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c6">h</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K63" title="MODELC_1:func.63">TransEU</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> implies for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font> iff <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:178_3"/>
 <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c6">h</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K63" title="MODELC_1:func.63">TransEU</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font> iff <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font> )</span><br/>

<a NAME="E2:178_3"/><b>then </b><i><font color="Green" title="E137">A11</font></i>: 
 <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:178"><i><font color="Green" title="E135">A1</font></i></a>, <a class="ref" href="abian.html#D4" title="ABIAN:def.4">ABIAN:def 4</a></i>;<br/>
<a NAME="E3:178_3"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font> iff <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font> iff <font color="Maroon" title="c9">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font> )</span><br/>

<b>thus </b><a NAME="E1:178_3_1"/>
( <font color="Maroon" title="c9">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font> implies <font color="Maroon" title="c9">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font> implies <font color="Maroon" title="c9">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="178_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:178_3_1_1"/>
<font color="Maroon" title="c9">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font></span><br/>

<a NAME="E2:178_3_1_1"/><b>then </b>
<font color="Maroon" title="c9">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="c6">h</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c10">t</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="E4:178_3_1_1"/><i><font color="Green" title="E138">A12</font></i>: 
<font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t</font>
 <b>and </b><br/><a NAME="E5:178_3_1_1"/><i><font color="Green" title="E139">A13</font></i>: 
<font color="Maroon" title="c10">t</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>
 <b>by </b><i><a class="txt" href="#E2:178_3"><i><font color="Green" title="E137">A11</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:178_3_1_1"/>
<font color="Maroon" title="c9">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>
 <b>by </b><i><a class="txt" href="#E4:178_3_1_1"><i><font color="Green" title="E138">A12</font></i></a>, <a class="txt" href="#E5:178_3_1_1"><i><font color="Green" title="E139">A13</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:178_3_1"/>
<font color="Maroon" title="c9">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font></span><br/>

<a NAME="E3:178_3_1"/><b>then </b>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c10">t</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="E5:178_3_1"/><i><font color="Green" title="E138">A14</font></i>: 
<font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t</font>
 <b>and </b><br/><a NAME="E6:178_3_1"/><i><font color="Green" title="E139">A15</font></i>: 
<font color="Maroon" title="c10">t</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font>
 <b>by </b><i><a class="txt" href="#E2:178_3"><i><font color="Green" title="E137">A11</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:178_3_1"/>
<font color="Maroon" title="c9">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font>
 <b>by </b><i><a class="txt" href="#E5:178_3_1"><i><font color="Green" title="E138">A14</font></i></a>, <a class="txt" href="#E6:178_3_1"><i><font color="Green" title="E139">A15</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:178_3"/>
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font> iff <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:178"/>
( ( for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font> iff <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K61" title="MODELC_1:func.61">Foax</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font> ) ) iff  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c6">h</font> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K63" title="MODELC_1:func.63">TransEU</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> )
 <b>by </b><i><a class="txt" href="#E2:178"><i><font color="Green" title="E136">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
