<?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/>









<b>thus </b><a NAME="E1:49"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">p</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> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="valuat_1.html#D12" target="_self">VALUAT_1:def 12</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1"/>
not <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">p</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> 
 ;<br/>

<a NAME="E2:49_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">p</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#T43">MARGREL1:43</a></i>;<br/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">p</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 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"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p5">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">p</font></span>)</span></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>
<b>by </b><i><a class="ref" href="qc_lang2.html#D2">QC_LANG2:def 2</a></i>
<br/>.= 
 <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"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p5">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">p</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></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</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="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p5">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">p</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></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="margrel1.html#K9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</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="margrel1.html#K10">'&amp;'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K9">'not'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p5">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">J</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E4:49_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> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</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="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> <span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">p</font></span>)</span>,<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>, <a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/>
<a NAME="E5:49_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> <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>  &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> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">p</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><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/>
<a NAME="E6:49_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> <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>  &amp; <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">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">p</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#T41">MARGREL1:41</a></i>;<br/>
<a NAME="E7:49_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> <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>  &amp; <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="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> <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="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:49_1"/>
contradiction
 ;<br/>


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


</div>
