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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#NM1" title="RELSET_1:NM.1">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#K31" title="MODELC_1:func.31">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#K49" title="MODELC_1:func.49">BASSModel</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#K60" title="MODELC_1:func.60">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#K52" title="MODELC_1:func.52">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#K62" title="MODELC_1:func.62">TransEU</a> (<font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">g</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="relset_1.html#NM1" title="RELSET_1:NM.1">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K31" title="MODELC_1:func.31">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#K49" title="MODELC_1:func.49">BASSModel</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#K60" title="MODELC_1:func.60">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#K52" title="MODELC_1:func.52">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#K62" title="MODELC_1:func.62">TransEU</a> (<font color="Olive" title="b2">f</font>,<font color="Olive" title="b3">g</font>) )</span><br/><span class="kw">let </span><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#K31" title="MODELC_1:func.31">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K49" title="MODELC_1:func.49">BASSModel</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#K60" title="MODELC_1:func.60">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#K52" title="MODELC_1:func.52">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#K62" title="MODELC_1:func.62">TransEU</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font>) )</span><br/><span class="kw">let </span><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#K49" title="MODELC_1:func.49">BASSModel</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K60" title="MODELC_1:func.60">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#K52" title="MODELC_1:func.52">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#K62" title="MODELC_1:func.62">TransEU</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>) )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c7">H</font> =  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c6">h</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">Q</font> =  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">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/>
<span class="lab"><font color="Green" title="E131">A1</font></span>: <span class="p1">(<span class="default"><a href="modelc_1.html#K62" title="MODELC_1:func.62">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#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c6">h</font></span>)</span> = 
 <a href="modelc_1.html#K61" title="MODELC_1:func.61">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#K52" title="MODELC_1:func.52">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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#D73" target="_self" title="MODELC_1:def.73">Def75</a></span>
<br/>.= 
 <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">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 class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T31" target="_self" title="MODELC_1:th.31">Th31</a></span>
;<br/>
<a NAME="E2:180"/><span class="lab"><font color="Green" title="E132">A2</font></span>: 
(  <a href="modelc_1.html#K52" title="MODELC_1:func.52">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#K62" title="MODELC_1:func.62">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#K60" title="MODELC_1:func.60">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="180_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:180_2"/>
 <a href="modelc_1.html#K52" title="MODELC_1:func.52">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#K62" title="MODELC_1:func.62">TransEU</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K60" title="MODELC_1:func.60">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="E2:180_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A3</font></span>: 
 <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:180"><span class="lab"><font color="Green" title="E131">A1</font></span></a>, <a class="ref" href="abian.html#D4" title="ABIAN:def.4">ABIAN:def 4</a></span>;<br/>
<a NAME="E3:180_2"/>
 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#K60" title="MODELC_1:func.60">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="180_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K60" title="MODELC_1:func.60">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/>

<span class="kw">thus </span><a NAME="E1:180_2_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#K60" title="MODELC_1:func.60">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K60" title="MODELC_1:func.60">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="180_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:180_2_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K60" title="MODELC_1:func.60">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="E2:180_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c6">h</font>
 
;<br/>
<a NAME="E3:180_2_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K60" title="MODELC_1:func.60">Foax</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:180_2"><span class="lab"><font color="Green" title="E133">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:180_2_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#K60" title="MODELC_1:func.60">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:180_2_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#K60" title="MODELC_1:func.60">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E3:180_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">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/>
<a NAME="E4:180_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c6">h</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:180_2"><span class="lab"><font color="Green" title="E133">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:180_2_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:180_2"/>
 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#K60" title="MODELC_1:func.60">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:180"/>
( (  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#K60" title="MODELC_1:func.60">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#K52" title="MODELC_1:func.52">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#K62" title="MODELC_1:func.62">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="180_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:180_3"/><span class="lab"><font color="Green" title="E133">A4</font></span>: 
 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#K60" title="MODELC_1:func.60">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="modelc_1.html#K52" title="MODELC_1:func.52">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#K62" title="MODELC_1:func.62">TransEU</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>)</span><br/>

<a NAME="E2:180_3"/><span class="lab"><font color="Green" title="E134">A5</font></span>: 
 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#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">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#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c6">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="180_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">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#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c6">h</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:180_3_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c6">h</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:180_3_1"/><span class="lab"><font color="Green" title="E135">A6</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font>
 <span class="kw">and </span><br/><a NAME="E4:180_3_1"/><span class="lab"><font color="Green" title="E136">A7</font></span>: 
<font color="Maroon" title="c10">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K60" title="MODELC_1:func.60">Foax</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>)
 ;<br/>
<a NAME="E5:180_3_1"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:180_3"><span class="lab"><font color="Green" title="E133">A4</font></span></a>, <a class="txt" href="#E4:180_3_1"><span class="lab"><font color="Green" title="E136">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:180_3_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c6">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:180_3_1"><span class="lab"><font color="Green" title="E135">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:180_3"/>
 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#K52" title="MODELC_1:func.52">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#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">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="180_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K52" title="MODELC_1:func.52">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#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">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/>

<span class="kw">assume </span><a NAME="E1:180_3_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c6">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">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/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:180_3_2"/><span class="lab"><font color="Green" title="E135">A8</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font>
 <span class="kw">and </span><br/><a NAME="E4:180_3_2"/><span class="lab"><font color="Green" title="E136">A9</font></span>: 
<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>
 ;<br/>
<a NAME="E5:180_3_2"/>
<font color="Maroon" title="c10">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K60" title="MODELC_1:func.60">Foax</a> (<font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c6">h</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:180_3"><span class="lab"><font color="Green" title="E133">A4</font></span></a>, <a class="txt" href="#E4:180_3_2"><span class="lab"><font color="Green" title="E136">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:180_3_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:180_3_2"><span class="lab"><font color="Green" title="E135">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:180_3"/><span class="kw">then </span>
 <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:180_3"><span class="lab"><font color="Green" title="E134">A5</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:180_3"/>
 <a href="modelc_1.html#K52" title="MODELC_1:func.52">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#K62" title="MODELC_1:func.62">TransEU</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:180"><span class="lab"><font color="Green" title="E131">A1</font></span></a>, <a class="ref" href="abian.html#D4" title="ABIAN:def.4">ABIAN:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:180"/>
( (  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#K60" title="MODELC_1:func.60">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#K52" title="MODELC_1:func.52">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#K62" title="MODELC_1:func.62">TransEU</a> (<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:180"><span class="lab"><font color="Green" title="E132">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
