<?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="Red">::  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#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> 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"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="knaster.html#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Olive" title="b3">f</font></span>)</span></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="Red">::  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#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> 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"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="knaster.html#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Olive" title="b2">f</font></span>)</span></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#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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</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"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="knaster.html#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Olive" title="b1">f</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="knaster.html#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">g</font> =  <a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">h</font> =  <a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <span class="p1">(<span class="default"><a href="knaster.html#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">S</font>,<span class="p2">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font>;<br/>
<a NAME="E1:170"/><span class="lab"><font color="Green" title="E129">A1</font></span>: 
 <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#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">S</font>,<span class="p3">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Maroon" title="c4">f</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#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T32" target="_self" title="MODELC_1:th.32">Th32</a></span>;<br/>
<a NAME="E2:170"/><span class="kw">then </span>
 <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#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">S</font>,<span class="p3">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="knaster.html#T7" title="KNASTER:th.7">KNASTER:7</a></span>;<br/>
<a NAME="E3:170"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A2</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>  <a href="modelc_1.html#K54" title="MODELC_1:func.54">Tau</a> <span class="p1">(<span class="default"><a href="knaster.html#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">S</font>,<span class="p2">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Maroon" title="c4">f</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#K55" title="MODELC_1:func.55">Fax</a> <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#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">S</font>,<span class="p3">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T42" target="_self" title="MODELC_1:th.42">Th42</a></span>;<br/>
<a NAME="E4:170"/><span class="lab"><font color="Green" title="E131">A3</font></span>: 
 <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#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">S</font>,<span class="p3">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</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#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="170_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">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#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">S</font>,<span class="p3">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font></span>)</span> or <font color="Maroon" title="c7">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#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:170_1"/>
<font color="Maroon" title="c7">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#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">S</font>,<span class="p3">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">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#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">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="E3:170_1"/><span class="lab"><font color="Green" title="E132">A4</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font>
 <span class="kw">and </span><br/><a NAME="E4:170_1"/><span class="lab"><font color="Green" title="E133">A5</font></span>: 
<font color="Maroon" title="c8">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#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">S</font>,<span class="p2">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">BASSIGN</font>
 ;<br/>
<a NAME="E5:170_1"/>
<font color="Maroon" title="c8">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:170"><span class="lab"><font color="Green" title="E130">A2</font></span></a>, <a class="txt" href="#E4:170_1"><span class="lab"><font color="Green" title="E133">A5</font></span></a>, <a class="ref" href="modelc_1.html#T41" target="_self" title="MODELC_1:th.41">Th41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:170_1"/>
<font color="Maroon" title="c7">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#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:170_1"><span class="lab"><font color="Green" title="E132">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:170"/>
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#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</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#K55" title="MODELC_1:func.55">Fax</a> <font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T39" target="_self" title="MODELC_1:th.39">Th39</a></span>;<br/>
<a NAME="E6:170"/><span class="kw">then </span>
 <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="abian.html#R2" title="ABIAN:pred.2">is_a_fixpoint_of</a>  <a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_1.html#T42" target="_self" title="MODELC_1:th.42">Th42</a></span>;<br/>
<a NAME="E7:170"/><span class="kw">then </span>
 <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="knaster.html#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="knaster.html#T10" title="KNASTER:th.10">KNASTER:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:170"/>
 <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K28" title="MODELC_1:func.28">EG</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="knaster.html#K2" title="KNASTER:func.2">gfp</a> <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K60" title="MODELC_1:func.60">TransEG</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:170"><span class="lab"><font color="Green" title="E129">A1</font></span></a>, <a class="txt" href="#E4:170"><span class="lab"><font color="Green" title="E131">A3</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
