<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">S</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be  <a href="partfun1.html#V1">total</a>  <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">S</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">BASSIGN</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K32" target="_self">ModelSP</a> <font color="Maroon">S</font></span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="modelc_1.html#NM2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" target="_self">CTLModel</a> <font color="Maroon">R</font>,<font color="Maroon">BASSIGN</font></span>)</span>, <font color="Maroon">g</font> be   <a href="modelc_1.html#NM2">Assign</a> of <span class="p1">(<span class="default"><a href="modelc_1.html#K50" target="_self">CTLModel</a> <font color="Maroon">R</font>,<font color="Maroon">BASSIGN</font></span>)</span>;<br/>









<b>set </b><font color="Maroon">V</font> =  <a href="modelc_1.html#K50" target="_self">CTLModel</a> <font color="Maroon">R</font>,<font color="Maroon">BASSIGN</font>;<br/>
<a NAME="E1:126"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">f</font> <a href="modelc_1.html#K26" target="_self">'&amp;'</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="modelc_1.html#K42" target="_self">And_0</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E25"><i><font color="Green">Lemma88</font></i></a></i>;<br/>
<a NAME="E2:126"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> <a href="modelc_1.html#K26" target="_self">'&amp;'</a> <font color="Maroon">g</font> implies ( <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="126_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:126_1"/>
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> <a href="modelc_1.html#K26" target="_self">'&amp;'</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E2:126_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <span class="p2">(<span class="default"><a href="modelc_1.html#K42" target="_self">And_0</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>,<font color="Maroon">S</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:126_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="modelc_1.html#K4" target="_self">Castboolean</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">f</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xboolean.html#K4">'&amp;'</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K4" target="_self">Castboolean</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">g</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="modelc_1.html#D18" target="_self">Def18</a></i>;<br/>
<a NAME="E4:126_1"/><b>then </b>
(  <a href="modelc_1.html#K4" target="_self">Castboolean</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">f</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp;  <a href="modelc_1.html#K4" target="_self">Castboolean</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">g</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 
<b>by </b><i><a class="ref" href="xboolean.html#T101">XBOOLEAN:101</a></i>;<br/>
<a NAME="E5:126_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">f</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">g</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:126_1"/>
( <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:126"/>
( <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> implies <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> <a href="modelc_1.html#K26" target="_self">'&amp;'</a> <font color="Maroon">g</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="126_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:126_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:126_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font>
 ;<br/>

<a NAME="E3:126_2"/>
( <span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">f</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">g</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:126_2"/><b>then </b>
(  <a href="modelc_1.html#K4" target="_self">Castboolean</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">f</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp;  <a href="modelc_1.html#K4" target="_self">Castboolean</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">g</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E5:126_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="modelc_1.html#K4" target="_self">Castboolean</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">f</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xboolean.html#K4">'&amp;'</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K4" target="_self">Castboolean</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <font color="Maroon">g</font>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
;<br/>
<a NAME="E6:126_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="modelc_1.html#K33" target="_self">Fid</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K26" target="_self">'&amp;'</a> <font color="Maroon">g</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i>, <a class="ref" href="modelc_1.html#D18" target="_self">Def18</a></i>;<br/>
<b>hence </b><a NAME="E7:126_2"/>
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> <a href="modelc_1.html#K26" target="_self">'&amp;'</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:126"/>
( <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> <a href="modelc_1.html#K26" target="_self">'&amp;'</a> <font color="Maroon">g</font> iff ( <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> ) )
 <b>by </b><i/>;<br/>


</div>
