<?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/>







<a NAME="E1:154"/><i><font color="Green" title="E116">A1</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">s</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#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c4">f</font></span>)</span> iff <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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> )
 
<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" title="c6">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:154_1"/><i><font color="Green" title="E117">A2</font></i>: 
( <font color="Maroon" title="c6">s</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#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c4">f</font></span>)</span> implies <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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> )
 
<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" title="c6">s</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#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x</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="E3:154_1_1"/><i><font color="Green" title="E118">A3</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font>
 <b>and </b><br/><a NAME="E4:154_1_1"/><i><font color="Green" title="E119">A4</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c4">f</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c6">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="modelc_1.html#E3:154_1_1"><i><font color="Green" title="E118">A3</font></i></a></i>;<br/>
<a NAME="E6:154_1_1"/>
not <font color="Maroon" title="c8">s</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>
 
<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" title="c8">s</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>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">y</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="E3:154_1_1_1"/><i><font color="Green" title="E120">A5</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font>
 <b>and </b><br/><a NAME="E4:154_1_1_1"/><i><font color="Green" title="E121">A6</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">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#E3:154_1_1"><i><font color="Green" title="E118">A3</font></i></a>, <a class="txt" href="modelc_1.html#E4:154_1_1"><i><font color="Green" title="E119">A4</font></i></a>, <a class="txt" href="modelc_1.html#E3:154_1_1_1"><i><font color="Green" title="E120">A5</font></i></a>, <a class="txt" href="modelc_1.html#E4:154_1_1_1"><i><font color="Green" title="E121">A6</font></i></a>, <a class="ref" href="modelc_1.html#T12" target="_self" title="MODELC_1:th.12">Th12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:154_1_1"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:154_1"/>
( <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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> implies <font color="Maroon" title="c6">s</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#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c4">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" title="E118">A7</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c6">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E3:154_1_2"/>
not <font color="Maroon" title="c7">s</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#E1:154_1_2"><i><font color="Green" title="E118">A7</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:154_1_2"/><b>then </b>
<font color="Maroon" title="c7">s</font> <a href="modelc_1.html#NR4" title="MODELC_1:NR.4">|/=</a> <font color="Maroon" title="c4">f</font>
 
;<br/>
<a NAME="E5:154_1_2"/><b>then </b>
<font color="Maroon" title="c7">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a>  <a href="modelc_1.html#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="modelc_1.html#T12" target="_self" title="MODELC_1:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E6:154_1_2"/>
<font color="Maroon" title="c6">s</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#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c4">f</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:154_1"/>
( <font color="Maroon" title="c6">s</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#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c4">f</font></span>)</span> iff <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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> )
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:154_1"><i><font color="Green" title="E117">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:154"/><i><font color="Green" title="E117">A8</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">s</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"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K26" title="MODELC_1:func.26">'&amp;'</a> <font color="Maroon" title="c5">g</font></span>)</span> iff <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c5">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" title="c6">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:154_2"/>
( <font color="Maroon" title="c6">s</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"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K26" title="MODELC_1:func.26">'&amp;'</a> <font color="Maroon" title="c5">g</font></span>)</span> implies <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c5">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" title="c6">s</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"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K26" title="MODELC_1:func.26">'&amp;'</a> <font color="Maroon" title="c5">g</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x</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="E3:154_2_1"/><i><font color="Green" title="E118">A9</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font>
 <b>and </b><br/><a NAME="E4:154_2_1"/><i><font color="Green" title="E119">A10</font></i>: 
<font color="Maroon" title="c7">x</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> <font color="Maroon" title="c5">g</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c6">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="modelc_1.html#E3:154_2_1"><i><font color="Green" title="E118">A9</font></i></a></i>;<br/>
<a NAME="E6:154_2_1"/><i><font color="Green" title="E120">A11</font></i>: 
( <font color="Maroon" title="c8">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c8">s</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#E3:154_2_1"><i><font color="Green" title="E118">A9</font></i></a>, <a class="txt" href="modelc_1.html#E4:154_2_1"><i><font color="Green" title="E119">A10</font></i></a>, <a class="ref" href="modelc_1.html#T13" target="_self" title="MODELC_1:th.13">Th13</a></i>;<br/>
<a NAME="E7:154_2_1"/><b>then </b><i><font color="Green" title="E121">A12</font></i>: 
<font color="Maroon" title="c8">s</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>
 
;<br/>
<a NAME="E8:154_2_1"/>
<font color="Maroon" title="c8">s</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="c5">g</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E6:154_2_1"><i><font color="Green" title="E120">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:154_2_1"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c5">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="modelc_1.html#E7:154_2_1"><i><font color="Green" title="E121">A12</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


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

<b>assume </b><a NAME="E2:154_2"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c5">g</font></span>)</span>
 ;<br/>

<a NAME="E3:154_2"/><b>then </b><i><font color="Green" title="E118">A13</font></i>: 
( <font color="Maroon" title="c6">s</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; <font color="Maroon" title="c6">s</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="c5">g</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">x</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="E5:154_2"/><i><font color="Green" title="E119">A14</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font>
 <b>and </b><br/><a NAME="E6:154_2"/><i><font color="Green" title="E120">A15</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c6">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="modelc_1.html#E5:154_2"><i><font color="Green" title="E119">A14</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">y</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="E9:154_2"/><i><font color="Green" title="E121">A16</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font>
 <b>and </b><br/><a NAME="E10:154_2"/><i><font color="Green" title="E122">A17</font></i>: 
<font color="Maroon" title="c9">y</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#E3:154_2"><i><font color="Green" title="E118">A13</font></i></a></i>;<br/>
<a NAME="E11:154_2"/>
<font color="Maroon" title="c8">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> <font color="Maroon" title="c5">g</font>
 
<b>by </b><i><a class="txt" href="modelc_1.html#E5:154_2"><i><font color="Green" title="E119">A14</font></i></a>, <a class="txt" href="modelc_1.html#E6:154_2"><i><font color="Green" title="E120">A15</font></i></a>, <a class="txt" href="modelc_1.html#E9:154_2"><i><font color="Green" title="E121">A16</font></i></a>, <a class="txt" href="modelc_1.html#E10:154_2"><i><font color="Green" title="E122">A17</font></i></a>, <a class="ref" href="modelc_1.html#T13" target="_self" title="MODELC_1:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E12:154_2"/>
<font color="Maroon" title="c6">s</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"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K26" title="MODELC_1:func.26">'&amp;'</a> <font color="Maroon" title="c5">g</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:154"/>
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">s</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"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">'or'</a> <font color="Maroon" title="c5">g</font></span>)</span> iff <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c5">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" title="c6">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:154_3"/><i><font color="Green" title="E118">A18</font></i>: 
( <font color="Maroon" title="c6">s</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"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">'or'</a> <font color="Maroon" title="c5">g</font></span>)</span> implies <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c5">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" title="c6">s</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"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">'or'</a> <font color="Maroon" title="c5">g</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x</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="E3:154_3_1"/><i><font color="Green" title="E119">A19</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font>
 <b>and </b><br/><a NAME="E4:154_3_1"/><i><font color="Green" title="E120">A20</font></i>: 
<font color="Maroon" title="c7">x</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#K30" title="MODELC_1:func.30">'or'</a> <font color="Maroon" title="c5">g</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c6">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="modelc_1.html#E3:154_3_1"><i><font color="Green" title="E119">A19</font></i></a></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" title="c8">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font> or <font color="Maroon" title="c8">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c5">g</font> )
 </span><b>by </b><i><a class="txt" href="modelc_1.html#E3:154_3_1"><i><font color="Green" title="E119">A19</font></i></a>, <a class="txt" href="modelc_1.html#E4:154_3_1"><i><font color="Green" title="E120">A20</font></i></a>, <a class="ref" href="modelc_1.html#T17" target="_self" title="MODELC_1:th.17">Th17</a></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" title="c8">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 ;<br/><div class="add"><a NAME="E2:154_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">s</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>
 ;<br/><b>hence </b><a NAME="E3:154_3_1_1_1"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c5">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c8">s</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c5">g</font>
 ;<br/><div class="add"><a NAME="E2:154_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">s</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="c5">g</font>
 ;<br/><b>hence </b><a NAME="E3:154_3_1_1_2"/>
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c5">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c5">g</font></span>)</span> implies <font color="Maroon" title="c6">s</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"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">'or'</a> <font color="Maroon" title="c5">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" title="E119">A21</font></i>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c5">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" title="c6">s</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> or <font color="Maroon" title="c6">s</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="c5">g</font> )
 </span><b>by </b><i><a class="txt" href="modelc_1.html#E1:154_3_2"><i><font color="Green" title="E119">A21</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c6">s</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>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</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="E3:154_3_2_1_1"/><i><font color="Green" title="E120">A22</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font>
 <b>and </b><br/><a NAME="E4:154_3_2_1_1"/><i><font color="Green" title="E121">A23</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c4">f</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c6">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="modelc_1.html#E3:154_3_2_1_1"><i><font color="Green" title="E120">A22</font></i></a></i>;<br/><a NAME="E6:154_3_2_1_1"/>
<font color="Maroon" title="c8">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#K30" title="MODELC_1:func.30">'or'</a> <font color="Maroon" title="c5">g</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:154_3_2_1_1"><i><font color="Green" title="E120">A22</font></i></a>, <a class="txt" href="modelc_1.html#E4:154_3_2_1_1"><i><font color="Green" title="E121">A23</font></i></a>, <a class="ref" href="modelc_1.html#T17" target="_self" title="MODELC_1:th.17">Th17</a></i>;<br/><b>hence </b><a NAME="E7:154_3_2_1_1"/>
<font color="Maroon" title="c6">s</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"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">'or'</a> <font color="Maroon" title="c5">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" title="c6">s</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="c5">g</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">x</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="E3:154_3_2_1_2"/><i><font color="Green" title="E120">A24</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font>
 <b>and </b><br/><a NAME="E4:154_3_2_1_2"/><i><font color="Green" title="E121">A25</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="modelc_1.html#R3" title="MODELC_1:pred.3">|=</a> <font color="Maroon" title="c5">g</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c6">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="modelc_1.html#E3:154_3_2_1_2"><i><font color="Green" title="E120">A24</font></i></a></i>;<br/><a NAME="E6:154_3_2_1_2"/>
<font color="Maroon" title="c8">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#K30" title="MODELC_1:func.30">'or'</a> <font color="Maroon" title="c5">g</font>
 <b>by </b><i><a class="txt" href="modelc_1.html#E3:154_3_2_1_2"><i><font color="Green" title="E120">A24</font></i></a>, <a class="txt" href="modelc_1.html#E4:154_3_2_1_2"><i><font color="Green" title="E121">A25</font></i></a>, <a class="ref" href="modelc_1.html#T17" target="_self" title="MODELC_1:th.17">Th17</a></i>;<br/><b>hence </b><a NAME="E7:154_3_2_1_2"/>
<font color="Maroon" title="c6">s</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"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">'or'</a> <font color="Maroon" title="c5">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" title="c6">s</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"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">'or'</a> <font color="Maroon" title="c5">g</font></span>)</span> iff <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c5">g</font></span>)</span> )
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:154_3"><i><font color="Green" title="E118">A18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:154"/>
(  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K25" title="MODELC_1:func.25">'not'</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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> &amp;  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K26" title="MODELC_1:func.26">'&amp;'</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c5">g</font></span>)</span> &amp;  <a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="modelc_1.html#K30" title="MODELC_1:func.30">'or'</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="modelc_1.html#K53" title="MODELC_1:func.53">SIGMA</a> <font color="Maroon" title="c5">g</font></span>)</span> )
 <b>by </b><i><a class="txt" href="modelc_1.html#E1:154"><i><font color="Green" title="E116">A1</font></i></a>, <a class="txt" href="modelc_1.html#E2:154"><i><font color="Green" title="E117">A8</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div>
