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<b>let </b><font color="Maroon">T</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">T</font> <a href="cqc_the1.html#NR1" target="_self">is_a_theory</a> 
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S</font> <a href="cqc_the1.html#NR1" target="_self">is_a_theory</a> 
 ;<br/>

<a NAME="E3:7"/>
(  <a href="cqc_lang.html#K9">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp;  <a href="cqc_lang.html#K9">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E4:7"/>
 <a href="cqc_lang.html#K9">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cqc_the1.html#D1" target="_self">CQC_THE1:def 1</a><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">r</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">s</font> be   <a href="qc_lang1.html#NM8">QC-formula</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/>











<a NAME="E5:7"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<a NAME="E7:7"/>
( <font color="Maroon">p</font> <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">p</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; <font color="Maroon">p</font> <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">p</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E8:7"/>
<font color="Maroon">p</font> <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">p</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<a NAME="E9:7"/>
( <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> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</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> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; <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> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</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> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E10:7"/>
<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> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</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> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<a NAME="E11:7"/>
( <span class="p1">(<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="p1">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; <span class="p1">(<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="p1">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E12:7"/>
<span class="p1">(<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="p1">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<a NAME="E13:7"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; <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> <font color="Maroon">T</font> implies <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> ) &amp; ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <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> <font color="Maroon">S</font> implies <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E14:7"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> &amp; <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> <font color="Maroon">T</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> implies <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<a NAME="E15:7"/>
( <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E16:7"/>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<a NAME="E17:7"/>
( ( <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> <font color="Maroon">T</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> implies <font color="Maroon">p</font> <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>,<font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> ) &amp; ( <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> <font color="Maroon">S</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> implies <font color="Maroon">p</font> <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>,<font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E18:7"/>
( <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> <font color="Maroon">T</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> implies <font color="Maroon">p</font> <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>,<font color="Maroon">q</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<a NAME="E19:7"/>
( ( <font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a>  &amp; <font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a>  &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> implies <font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> ) &amp; ( <font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a>  &amp; <font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a>  &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> implies <font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> ) )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E20:7"/>
( <font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a>  &amp; <font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a>  &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> implies <font color="Maroon">s</font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div>
