<?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> 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#K27" title="MODELC_1:func.27">EX</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">Pred</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Olive" title="b3">f</font></span>)</span>,<font color="Olive" title="b1">R</font></span><br/><b>let </b><font color="Maroon" title="c2">R</font> be  <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">BASSIGN</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span><br/> for <font color="Olive" title="b2">f</font> 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#K27" title="MODELC_1:func.27">EX</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">Pred</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Olive" title="b2">f</font></span>)</span>,<font color="Maroon" title="c2">R</font></span><br/><b>let </b><font color="Maroon" title="c3">BASSIGN</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" title="MODELC_1:func.32">ModelSP</a> <font color="Maroon" title="c1">S</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> 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#K27" title="MODELC_1:func.27">EX</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">Pred</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Olive" title="b1">f</font></span>)</span>,<font color="Maroon" title="c2">R</font></span><br/><b>let </b><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:()"><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"><a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">Pred</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font></span><br/>







<b>set </b><font color="Maroon" title="c5">g</font> =  <a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c4">f</font>;<br/>
<b>set </b><font color="Maroon" title="c6">H</font> =  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font>;<br/>
<a NAME="E1:180"/><i><font color="Green" title="E136">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c4">f</font></span>)</span> 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">Pred</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="180_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="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#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c4">f</font></span>)</span> implies <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">Pred</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font> )</span><br/>

<b>assume </b><a NAME="E1:180_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#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c4">f</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="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">Pred</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font></span><br/>

<b>then consider </b><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><b> such that </b><br/><a NAME="E3:180_1"/><i><font color="Green" title="E137">A2</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font>
 <b>and </b><br/><a NAME="E4:180_1"/><i><font color="Green" title="E138">A3</font></i>: 
<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">EX</a> <font color="Maroon" title="c4">f</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c9">s1</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="E6:180_1"/><i><font color="Green" title="E139">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c9">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>and </b><br/><a NAME="E7:180_1"/><i><font color="Green" title="E140">A5</font></i>: 
<font color="Maroon" title="c9">s1</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E4:180_1"><i><font color="Green" title="E138">A3</font></i></a>, <a class="ref" href="modelc_1.html#T28" target="_self" title="MODELC_1:th.28">Th28</a></i>;<br/>
<a NAME="E8:180_1"/>
<font color="Maroon" title="c9">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E7:180_1"><i><font color="Green" title="E140">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:180_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">Pred</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:180_1"><i><font color="Green" title="E137">A2</font></i></a>, <a class="txt" href="modelc_1.html#E6:180_1"><i><font color="Green" title="E139">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>
<a NAME="E2:180"/>
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">Pred</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="180_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="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">Pred</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font> implies <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#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c4">f</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:180_2"/>
<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">Pred</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c4">f</font></span>)</span></span><br/>

<b>then consider </b><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><b> such that </b><br/><a NAME="E3:180_2"/><i><font color="Green" title="E137">A6</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font>
 <b>and </b><br/><a NAME="E4:180_2"/><i><font color="Green" title="E138">A7</font></i>: 
 ex <font color="Olive" title="b1">s1</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="Olive" title="b1">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">s</font>,<font color="Olive" title="b1">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c9">s1</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="E6:180_2"/><i><font color="Green" title="E139">A8</font></i>: 
<font color="Maroon" title="c9">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E7:180_2"/><i><font color="Green" title="E140">A9</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">s</font>,<font color="Maroon" title="c9">s1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E4:180_2"><i><font color="Green" title="E138">A7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">s2</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="E9:180_2"/><i><font color="Green" title="E141">A10</font></i>: 
<font color="Maroon" title="c9">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s2</font>
 <b>and </b><br/><a NAME="E10:180_2"/><i><font color="Green" title="E142">A11</font></i>: 
<font color="Maroon" title="c10">s2</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E6:180_2"><i><font color="Green" title="E139">A8</font></i></a></i>;<br/>
<a NAME="E11:180_2"/>
<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">EX</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E7:180_2"><i><font color="Green" title="E140">A9</font></i></a>, <a class="txt" href="modelc_1.html#E9:180_2"><i><font color="Green" title="E141">A10</font></i></a>, <a class="txt" href="modelc_1.html#E10:180_2"><i><font color="Green" title="E142">A11</font></i></a>, <a class="ref" href="modelc_1.html#T28" target="_self" title="MODELC_1:th.28">Th28</a></i>;<br/>
<b>hence </b><a NAME="E12:180_2"/>
<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#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c4">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:180_2"><i><font color="Green" title="E137">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:180"/>
 <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="modelc_1.html#K52" title="MODELC_1:func.52">Pred</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:180"><i><font color="Green" title="E136">A1</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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