<?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">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/>







<a NAME="E1:154"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K25" target="_self">'not'</a> <font color="Maroon">f</font></span>)</span> iff <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:154_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K25" target="_self">'not'</a> <font color="Maroon">f</font></span>)</span> implies <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:154_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K25" target="_self">'not'</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:154_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:154_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K25" target="_self">'not'</a> <font color="Maroon">f</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E6:154_1_1"/>
not <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:154_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:154_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:154_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">y</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font>
 ;<br/>
<b>thus </b><a NAME="E5:154_1_1_1"/>
contradiction
 <b>by </b><i>, , , , <a class="txt" href="modelc_1.html#E33"><i><font color="Green">Lemma98</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:154_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:154_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> implies <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K25" target="_self">'not'</a> <font color="Maroon">f</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:154_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E3:154_1_2"/>
not <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:154_1_2"/><b>then </b>
<font color="Maroon">s</font> <a href="modelc_1.html#NR4" target="_self">|/=</a> <font color="Maroon">f</font>
 
;<br/>
<a NAME="E5:154_1_2"/><b>then </b>
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a>  <a href="modelc_1.html#K25" target="_self">'not'</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E33"><i><font color="Green">Lemma98</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:154_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K25" target="_self">'not'</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:154_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K25" target="_self">'not'</a> <font color="Maroon">f</font></span>)</span> iff <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:154"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<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> iff <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:154_2"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<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> implies <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:154_2_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<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>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:154_2_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:154_2_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x</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/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E6:154_2_1"/><i><font color="Green">E59</font></i>: 
( <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>, , </i>;<br/>
<a NAME="E7:154_2_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font>
 
;<br/>
<a NAME="E8:154_2_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:154_2_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:154_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<a NAME="E3:154_2"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E5:154_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E6:154_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E9:154_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E10:154_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">y</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>
<a NAME="E11:154_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>, , , , </i>;<br/>
<b>hence </b><a NAME="E12:154_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<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>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:154"/>
for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K30" target="_self">'or'</a> <font color="Maroon">g</font></span>)</span> iff <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:154_3"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K30" target="_self">'or'</a> <font color="Maroon">g</font></span>)</span> implies <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:154_3_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K30" target="_self">'or'</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:154_3_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:154_3_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> <a href="modelc_1.html#K30" target="_self">'or'</a> <font color="Maroon">g</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="154_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:154_3_1_1"/>
( <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font> or <font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font> )
 </span><b>by </b><i>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="154_3_1_1_1"><b>suppose </b></a><a NAME="E1:154_3_1_1_1"/>
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E2:154_3_1_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font>
 ;<br/><b>hence </b><a NAME="E3:154_3_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="154_3_1_1_2"><b>suppose </b></a><a NAME="E1:154_3_1_1_2"/>
<font color="Maroon">s</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E2:154_3_1_1_2"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font>
 ;<br/><b>hence </b><a NAME="E3:154_3_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:154_3"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> implies <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K30" target="_self">'or'</a> <font color="Maroon">g</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:154_3_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="154_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:154_3_2_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font> or <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="154_3_2_1_1"><b>suppose </b></a><a NAME="E1:154_3_2_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:154_3_2_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:154_3_2_1_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">x</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">f</font>
 ;<br/><b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E6:154_3_2_1_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#K30" target="_self">'or'</a> <font color="Maroon">g</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E7:154_3_2_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K30" target="_self">'or'</a> <font color="Maroon">g</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="154_3_2_1_2"><b>suppose </b></a><a NAME="E1:154_3_2_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:154_3_2_1_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:154_3_2_1_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">x</font> <a href="modelc_1.html#R3" target="_self">|=</a> <font color="Maroon">g</font>
 ;<br/><b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E6:154_3_2_1_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#K30" target="_self">'or'</a> <font color="Maroon">g</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E7:154_3_2_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K30" target="_self">'or'</a> <font color="Maroon">g</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:154_3"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K30" target="_self">'or'</a> <font color="Maroon">g</font></span>)</span> iff <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:154"/>
(  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K25" target="_self">'not'</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> &amp;  <a href="modelc_1.html#K53" target="_self">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="modelc_1.html#K30" target="_self">'or'</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" target="_self">SIGMA</a> <font color="Maroon">g</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a>, </i>;<br/>


</div>
