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<b>let </b><font color="Maroon" title="c1">Y</font>, <font color="Maroon" title="c2">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">CQC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:51"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">CQC-WFF</a>  :  ex <font color="Olive" title="b2">f</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">CQC-WFF</a> ,<a href="cqc_the1.html#K2" title="CQC_THE1:func.2">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive" title="b2">f</font> <a href="cqc_the1.html#R2" title="CQC_THE1:pred.2">is_a_proof_wrt</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="cqc_the1.html#K3" title="CQC_THE1:func.3">Effect</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> ) </span>}</span> 
 ;<br/>

<b>hence </b><a NAME="E2:51"/>
 <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">VERUM</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="cqc_the1.html#E77"><i><font color="Green" title="E35">Lm4</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cqc_the1.html#D1" target="_self" title="CQC_THE1:def.1">CQC_THE1:def 1</a><br/>

<b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font>, <font color="Maroon" title="c5">r</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon" title="c6">s</font> be   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>;<br/><b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>;<br/>











<b>thus </b><a NAME="E3:51"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10" title="CQC_LANG:func.10">'not'</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">=&gt;</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">=&gt;</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:51"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="cqc_the1.html#E78"><i><font color="Green" title="E36">Lm5</font></i></a></i>;<br/>

<b>thus </b><a NAME="E4:51"/>
<font color="Maroon" title="c3">p</font> <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10" title="CQC_LANG:func.10">'not'</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">=&gt;</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:51"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="cqc_the1.html#E79"><i><font color="Green" title="E37">Lm6</font></i></a></i>;<br/>

<b>thus </b><a NAME="E5:51"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">=&gt;</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10" title="CQC_LANG:func.10">'not'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">'&amp;'</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10" title="CQC_LANG:func.10">'not'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">'&amp;'</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:51"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="cqc_the1.html#E80"><i><font color="Green" title="E38">Lm7</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:51"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">'&amp;'</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">'&amp;'</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:51"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="cqc_the1.html#E81"><i><font color="Green" title="E39">Lm8</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:51"/>
( <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c3">p</font> <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">=&gt;</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> implies <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> )
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:51"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="cqc_the1.html#E82"><i><font color="Green" title="E40">Lm9</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:51"/>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">p</font></span>)</span> <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">=&gt;</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:51"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="cqc_the1.html#E83"><i><font color="Green" title="E41">Lm10</font></i></a></i>;<br/>

<b>thus </b><a NAME="E9:51"/>
( <font color="Maroon" title="c3">p</font> <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">=&gt;</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> &amp; not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c3">p</font> implies <font color="Maroon" title="c3">p</font> <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> )
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:51"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="cqc_the1.html#E84"><i><font color="Green" title="E42">Lm11</font></i></a></i>;<br/>

<b>thus </b><a NAME="E10:51"/>
( <font color="Maroon" title="c6">s</font> <a href="cqc_lang.html#K17" title="CQC_LANG:func.17">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">CQC-WFF</a>  &amp; <font color="Maroon" title="c6">s</font> <a href="cqc_lang.html#K17" title="CQC_LANG:func.17">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">CQC-WFF</a>  &amp; not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c6">s</font> <a href="cqc_lang.html#K17" title="CQC_LANG:func.17">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> implies <font color="Maroon" title="c6">s</font> <a href="cqc_lang.html#K17" title="CQC_LANG:func.17">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">Y</font> )
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:51"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="cqc_the1.html#E85"><i><font color="Green" title="E43">Lm12</font></i></a></i>;<br/>


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