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<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">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/>





<a NAME="E1:101"/>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<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></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#K11">'&amp;'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 
<b>by </b><i><a class="txt" href="cqc_the2.html#E17"><i><font color="Green">Lemma53</font></i></a></i>;<br/>
<a NAME="E2:101"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</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#K15">All</a> <font color="Maroon">x</font>,<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></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<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></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 
<b>by </b><i><a class="ref" href="lukasi_1.html#T63">LUKASI_1:63</a></i>;<br/>
<a NAME="E3:101"/>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="cqc_lang.html#K14">&lt;=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">q</font></span>)</span></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 
<b>by </b><i><a class="txt" href="cqc_the2.html#E14"><i><font color="Green">Lemma50</font></i></a></i>;<br/>
<a NAME="E4:101"/><b>then </b>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">q</font></span>)</span></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:101"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></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#K11">'&amp;'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:101"/><b>then </b>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</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> <span class="p4">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</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#K15">All</a> <font color="Maroon">x</font>,<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></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 
<b>by </b><i><a class="ref" href="lukasi_1.html#T63">LUKASI_1:63</a></i>;<br/>
<a NAME="E7:101"/><b>then </b>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</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> <span class="p4">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<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></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 
<b>by </b><i>, <a class="ref" href="lukasi_1.html#T43">LUKASI_1:43</a></i>;<br/>
<a NAME="E8:101"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">q</font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<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></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 
<b>by </b><i><a class="ref" href="qc_lang2.html#D2">QC_LANG2:def 2</a></i>;<br/>
<a NAME="E9:101"/>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</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></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#K11">'&amp;'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 
<b>by </b><i><a class="ref" href="lukasi_1.html#T65">LUKASI_1:65</a></i>;<br/>
<a NAME="E10:101"/><b>then </b>
(  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</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></span>)</span> is <a href="cqc_the1.html#V2">valid</a> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p0">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</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></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#K15">All</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p0">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<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></span>)</span> is <a href="cqc_the1.html#V2">valid</a> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:101"/><b>then </b>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p5">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<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></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 
<b>by </b><i><a class="ref" href="cqc_the1.html#T104">CQC_THE1:104</a></i>;<br/>
<a NAME="E12:101"/><b>then </b>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<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></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p0">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 
<b>by </b><i><a class="ref" href="lukasi_1.html#T63">LUKASI_1:63</a></i>;<br/>
<a NAME="E13:101"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">q</font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p5">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p0">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 
<b>by </b><i>, <a class="ref" href="lukasi_1.html#T43">LUKASI_1:43</a></i>;<br/>
<a NAME="E14:101"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">q</font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span></span>)</span> is <a href="cqc_the1.html#V2">valid</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="E15:101"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">q</font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<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></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 <b>by </b><i><a class="ref" href="qc_lang2.html#D5">QC_LANG2:def 5</a></i>;<br/>


</div>
