<?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> 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   <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K27" title="MODELC_1:func.27">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#K59" title="MODELC_1:func.59">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="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> 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   <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K27" title="MODELC_1:func.27">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#K59" title="MODELC_1:func.59">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#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> 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   <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K27" title="MODELC_1:func.27">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#K59" title="MODELC_1:func.59">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#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">  <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K27" title="MODELC_1:func.27">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#K59" title="MODELC_1:func.59">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#K27" title="MODELC_1:func.27">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#K53" title="MODELC_1:func.53">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#K59" title="MODELC_1:func.59">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="E1:172"/><span class="lab"><font color="Green" title="E125">A1</font></span>: 
 <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">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#K59" title="MODELC_1:func.59">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#K59" title="MODELC_1:func.59">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="E2:172"/><span class="kw">then </span>
 <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">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#K59" title="MODELC_1:func.59">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#K59" title="MODELC_1:func.59">TransEG</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="knaster.html#T5" title="KNASTER:th.5">KNASTER:5</a></span>;<br/>
<a NAME="E3:172"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">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#K53" title="MODELC_1:func.53">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#K59" title="MODELC_1:func.59">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#K54" title="MODELC_1:func.54">Fax</a> (<font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">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#K59" title="MODELC_1:func.59">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="E4:172"/><span class="lab"><font color="Green" title="E127">A3</font></span>: 
 <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">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#K59" title="MODELC_1:func.59">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#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K27" title="MODELC_1:func.27">EG</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="172_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="firebrick">:: 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="firebrick">::  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#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">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#K59" title="MODELC_1:func.59">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#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K27" title="MODELC_1:func.27">EG</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:172_1"/>
<font color="Maroon" title="c7">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#K53" title="MODELC_1:func.53">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#K59" title="MODELC_1:func.59">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="firebrick">::  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#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K27" title="MODELC_1:func.27">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="E3:172_1"/><span class="lab"><font color="Green" title="E128">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="E4:172_1"/><span class="lab"><font color="Green" title="E129">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#K53" title="MODELC_1:func.53">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#K59" title="MODELC_1:func.59">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="E5:172_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#K27" title="MODELC_1:func.27">EG</a> <font color="Maroon" title="c4">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:172"><span class="lab"><font color="Green" title="E126">A2</font></span></a>, <a class="txt" href="#E4:172_1"><span class="lab"><font color="Green" title="E129">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="E6:172_1"/>
<font color="Maroon" title="c7">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#K27" title="MODELC_1:func.27">EG</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:172_1"><span class="lab"><font color="Green" title="E128">A4</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="E5:172"/>
 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#K27" title="MODELC_1:func.27">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#K54" title="MODELC_1:func.54">Fax</a> (<font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><a href="modelc_1.html#K27" title="MODELC_1:func.27">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="E6:172"/><span class="kw">then </span>
 <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K27" title="MODELC_1:func.27">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#K59" title="MODELC_1:func.59">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="E7:172"/><span class="kw">then </span>
 <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K27" title="MODELC_1:func.27">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#K59" title="MODELC_1:func.59">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#T8" title="KNASTER:th.8">KNASTER:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:172"/>
 <a href="modelc_1.html#K52" title="MODELC_1:func.52">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K27" title="MODELC_1:func.27">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#K59" title="MODELC_1:func.59">TransEG</a> <font color="Maroon" title="c4">f</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:172"><span class="lab"><font color="Green" title="E125">A1</font></span></a>, <a class="txt" href="#E4:172"><span class="lab"><font color="Green" title="E127">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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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