<?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> 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   <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Olive" title="b4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><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>)</span></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> 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   <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Olive" title="b3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><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>)</span></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> 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   <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Olive" title="b2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><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>)</span></span><br/><b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</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">  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">S</font>,<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></span><br/>







<b>set </b><font color="Maroon" title="c6">h</font> = <font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font>;<br/>
<a NAME="E1:179"/>
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="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</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>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="modelc_1.html#T46" target="_self" title="MODELC_1:th.46">Th46</a></i>;<br/>
<a NAME="E2:179"/><b>then </b>
 <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span> <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="ref" href="modelc_1.html#T48" target="_self" title="MODELC_1:th.48">Th48</a></i>;<br/>
<a NAME="E3:179"/><b>then </b><i><font color="Green" title="E135">A1</font></i>: 
 <a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">S</font>,<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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="knaster.html#T10" title="KNASTER:th.10">KNASTER:10</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">p</font> =  <a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <span class="p1">(<span class="default"><a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">S</font>,<span class="p2">(<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></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font>;<br/>
<a NAME="E4:179"/><i><font color="Green" title="E136">A2</font></i>: 
 <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <span class="p2">(<span class="default"><a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">S</font>,<span class="p3">(<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></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">S</font>,<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>
 
<b>by </b><i><a class="ref" href="modelc_1.html#T32" target="_self" title="MODELC_1:th.32">Th32</a></i>;<br/>
<a NAME="E5:179"/>
 <a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">S</font>,<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="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="ref" href="knaster.html#T6" title="KNASTER:th.6">KNASTER:6</a></i>;<br/>
<a NAME="E6:179"/><b>then </b><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>  <a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <span class="p1">(<span class="default"><a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">S</font>,<span class="p2">(<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></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</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>,<span class="p1">(<span class="default"><a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <span class="p2">(<span class="default"><a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">S</font>,<span class="p3">(<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></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="modelc_1.html#E4:179"><i><font color="Green" title="E136">A2</font></i></a>, <a class="ref" href="modelc_1.html#T48" target="_self" title="MODELC_1:th.48">Th48</a></i>;<br/>
<a NAME="E7:179"/>
 <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K54" title="MODELC_1:func.54">Tau</a> <span class="p2">(<span class="default"><a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">S</font>,<span class="p3">(<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></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">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" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">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"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span> or <font color="Maroon" title="c8">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#K54" title="MODELC_1:func.54">Tau</a> <span class="p2">(<span class="default"><a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">S</font>,<span class="p3">(<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></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:179_1"/>
<font color="Maroon" title="c8">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"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</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="c8">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#K54" title="MODELC_1:func.54">Tau</a> <span class="p2">(<span class="default"><a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">S</font>,<span class="p3">(<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></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c9">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="E3:179_1"/><i><font color="Green" title="E138">A4</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">s</font>
 <b>and </b><br/><a NAME="E4:179_1"/><i><font color="Green" title="E139">A5</font></i>: 
<font color="Maroon" title="c9">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font>
 ;<br/>
<a NAME="E5:179_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#K54" title="MODELC_1:func.54">Tau</a> <span class="p1">(<span class="default"><a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">S</font>,<span class="p2">(<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></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E6:179"><i><font color="Green" title="E137">A3</font></i></a>, <a class="txt" href="modelc_1.html#E4:179_1"><i><font color="Green" title="E139">A5</font></i></a>, <a class="ref" href="modelc_1.html#T47" target="_self" title="MODELC_1:th.47">Th47</a></i>;<br/>
<b>hence </b><a NAME="E6:179_1"/>
<font color="Maroon" title="c8">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#K54" title="MODELC_1:func.54">Tau</a> <span class="p2">(<span class="default"><a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">S</font>,<span class="p3">(<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></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:179_1"><i><font color="Green" title="E138">A4</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="E8:179"/>
 <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K29" title="MODELC_1:func.29">EU</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="knaster.html#K1" title="KNASTER:func.1">lfp</a> <font color="Maroon" title="c1">S</font>,<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>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:179"><i><font color="Green" title="E135">A1</font></i></a>, <a class="txt" href="modelc_1.html#E4:179"><i><font color="Green" title="E136">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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>
