<?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> ;<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>;<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>;<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>;<br/>







<b>assume </b><a NAME="E1:167"/><i><font color="Green" title="E127">A1</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> <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#K55" title="MODELC_1:func.55">Fax</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font> )
 ;<br/>

<a NAME="E2:167"/><i><font color="Green" title="E128">A2</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>  st <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c5">g</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">EX</a> <font color="Maroon" title="c5">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="167_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>

<b>assume </b><a NAME="E1:167_1"/>
<font color="Maroon" title="c6">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c5">g</font>
 ;<br/>

<a NAME="E2:167_1"/><b>then </b>
<font color="Maroon" title="c6">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#K26" title="MODELC_1:func.26">'&amp;'</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c5">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:167"><i><font color="Green" title="E127">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:167_1"/>
<font color="Maroon" title="c6">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="c5">g</font>
 <b>by </b><i><a class="ref" href="modelc_1.html#T13" target="_self" title="MODELC_1:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:167"/>
for <font color="Olive" title="b1">s0</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">s0</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c5">g</font> holds <br/><font color="Olive" title="b1">s0</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="167_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">s0</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>

<b>assume </b><a NAME="E1:167_2"/>
<font color="Maroon" title="c6">s0</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c5">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">pai</font> being    <a href="modelc_1.html#M1" title="MODELC_1:mode.1">inf_path</a> of <font color="Maroon" title="c2">R</font><b> such that </b><br/><a NAME="E3:167_2"/><i><font color="Green" title="E129">A3</font></i>: 
<font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s0</font>
 <b>and </b><br/><a NAME="E4:167_2"/><i><font color="Green" title="E130">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c5">g</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E2:167"><i><font color="Green" title="E128">A2</font></i></a>, <a class="ref" href="modelc_1.html#T40" target="_self" title="MODELC_1:th.40">Th40</a></i>;<br/>
<a NAME="E5:167_2"/>
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="167_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:167_2_1"/>
<font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c5">g</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E4:167_2"><i><font color="Green" title="E130">A4</font></i></a></i>;<br/>
<a NAME="E2:167_2_1"/><b>then </b>
<font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</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#K26" title="MODELC_1:func.26">'&amp;'</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K27" title="MODELC_1:func.27">EX</a> <font color="Maroon" title="c5">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E1:167"><i><font color="Green" title="E127">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:167_2_1"/>
<font color="Maroon" title="c7">pai</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">n</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="ref" href="modelc_1.html#T13" target="_self" title="MODELC_1:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:167_2"/>
<font color="Maroon" title="c6">s0</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>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:167_2"><i><font color="Green" title="E129">A3</font></i></a>, <a class="ref" href="modelc_1.html#T15" target="_self" title="MODELC_1:th.15">Th15</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:167"/>
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>  st <font color="Olive" title="b1">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c5">g</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>
 ;<br/>


</div>
