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

<b>let </b><font color="Maroon">A</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">v</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">J</font> be    <a href="valuat_1.html#M1" target="_self">interpretation</a> of <font color="Maroon">A</font>;<br/>









<div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1" target="_self">|=</a> <font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E2:37_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><a NAME="E3:37_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="qc_lang2.html#D2">QC_LANG2:def 2</a></i>;<br/><a NAME="E4:37_1"/><b>then </b>
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><a NAME="E5:37_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/><a NAME="E6:37_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">p</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><a NAME="E7:37_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">p</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">q</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><a NAME="E8:37_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">p</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">q</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 <b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/><b>hence </b><a NAME="E9:37_1"/>
( <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">p</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">q</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 <b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_2"/><i><font color="Green">E30</font></i>: 
( <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">p</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">q</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 ;<br/><div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_2_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">p</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E2:37_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">p</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/><a NAME="E3:37_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><a NAME="E4:37_2_1"/><b>then </b>
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/><a NAME="E5:37_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><a NAME="E6:37_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="qc_lang2.html#D2">QC_LANG2:def 2</a></i>;<br/><b>hence </b><a NAME="E7:37_2_1"/>
<font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1" target="_self">|=</a> <font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_2_2"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">q</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><b>assume </b><a NAME="E2:37_2_2"/>
not <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1" target="_self">|=</a> <font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E3:37_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><a NAME="E4:37_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/><a NAME="E5:37_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="qc_lang2.html#D2">QC_LANG2:def 2</a></i>;<br/><a NAME="E6:37_2_2"/><b>then </b>
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i/>;<br/><a NAME="E7:37_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/><a NAME="E8:37_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">p</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><a NAME="E9:37_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">p</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="margrel1.html#K10">'&amp;'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">q</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i/>;<br/><a NAME="E10:37_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">p</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp;  <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">q</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 <b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/><b>hence </b><a NAME="E11:37_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="valuat_1.html#D3" target="_self">Def3</a>, <a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:37_2"/>
<font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1" target="_self">|=</a> <font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:37"/>
( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1" target="_self">|=</a> <font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font> iff ( <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">p</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  or <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">q</font>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  ) )
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:37"><i><font color="Green">E25</font></i></a></i>;<br/>


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