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

<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>thus </b><a NAME="E1:58"/>
( <font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K4">TAUT</a>  implies <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K4">TAUT</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<a NAME="E1:58_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K4">TAUT</a> 
 
<b>by </b><i><a class="txt" href="lukasi_1.html#E31"><i><font color="Green">Lemma46</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:58_1"/>
( <font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K4">TAUT</a>  implies <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K4">TAUT</a>  )
 <b>by </b><i><a class="ref" href="cqc_the1.html#T82">CQC_THE1:82</a></i>;<br/>


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

<b>thus </b><a NAME="E2:58"/>
( <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K4">TAUT</a>  implies <font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K4">TAUT</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>proof </b></a><div class="add">

<a NAME="E1:58_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K4">TAUT</a> 
 
<b>by </b><i><a class="ref" href="lukasi_1.html#T10" target="_self">Th10</a></i>;<br/>
<b>hence </b><a NAME="E2:58_2"/>
( <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K4">TAUT</a>  implies <font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="cqc_the1.html#K4">TAUT</a>  )
 <b>by </b><i><a class="ref" href="cqc_the1.html#T82">CQC_THE1:82</a></i>;<br/>


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


</div>
