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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/>





<b>assume </b><a NAME="E1:84"/><i><font color="Green">E57</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:84"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>1</sub></font> is <a href="cqc_the1.html#V2">valid</a>
 
<b>by </b><i><a class="ref" href="lukasi_1.html#T44">LUKASI_1:44</a></i>;<br/>
<a NAME="E3:84"/><b>then </b>
<font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 
<b>by </b><i><a class="txt" href="cqc_the2.html#E1:84"><i><font color="Green">E57</font></i></a>, <a class="ref" href="cqc_the1.html#T106">CQC_THE1:106</a></i>;<br/>
<a NAME="E4:84"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></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">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K13">'or'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> is <a href="cqc_the1.html#V2">valid</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K13">'or'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></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">Lemma14</font></i></a>, <a class="ref" href="cqc_the2.html#T43" target="_self">Th43</a></i>;<br/>
<b>hence </b><a NAME="E5:84"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K13">'or'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></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">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K13">'or'</a> <font color="Maroon">c<sub>2</sub></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#T43">LUKASI_1:43</a></i>;<br/>

<a NAME="E6:84"/>
( <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K13">'or'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K13">'or'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="cqc_the1.html#V2">valid</a> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K13">'or'</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="cqc_the1.html#V2">valid</a> )
 
<b>by </b><i><a class="txt" href="cqc_the2.html#E15"><i><font color="Green">Lemma15</font></i></a>, <a class="ref" href="cqc_the1.html#T105">CQC_THE1:105</a></i>;<br/>
<a NAME="E7:84"/><b>then </b>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K13">'or'</a> <font color="Maroon">c<sub>2</sub></font></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#K10">'not'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></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="E8:84"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K13">'or'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>2</sub></font> is <a href="cqc_the1.html#V2">valid</a>
 
<b>by </b><i><a class="ref" href="cqc_the2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E9:84"/>
( not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K13">'or'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="cqc_the2.html#E1:84"><i><font color="Green">E57</font></i></a>, <a class="ref" href="cqc_the2.html#T5" target="_self">Th5</a>, <a class="ref" href="qc_lang3.html#T11">QC_LANG3:11</a></i>;<br/>
<a NAME="E10:84"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K13">'or'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <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">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="cqc_the2.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E11:84"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K13">'or'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <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">c<sub>1</sub></font></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 
<b>by </b><i><a class="txt" href="cqc_the2.html#E8:84"><i><font color="Green">E58</font></i></a>, <a class="ref" href="cqc_the1.html#T106">CQC_THE1:106</a></i>;<br/>
<a NAME="E12:84"/><b>then </b>
( <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K13">'or'</a> <font color="Maroon">c<sub>2</sub></font></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#K10">'not'</a> <font color="Maroon">c<sub>1</sub></font></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> is <a href="cqc_the1.html#V2">valid</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>1</sub></font></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K13">'or'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></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#E16"><i><font color="Green">Lemma16</font></i></a>, <a class="ref" href="cqc_the2.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E13:84"/>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K13">'or'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K13">'or'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></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#T43">LUKASI_1:43</a></i>;<br/>


</div>
