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





<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="substut2.html#K2" target="_self">[</a><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>,<span class="p2">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span>;<br/>
<a NAME="E1:24"/><i><font color="Green">E15</font></i>: 
<span class="p1"><a href="substut2.html#K2" target="_self">[</a><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>,<span class="p2">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span> <a href="hidden.html#R1">=</a>  <a href="sublemma.html#K6">Sub_not</a> <span class="p1"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span>
 
<b>by </b><i><a class="ref" href="substut2.html#T16" target="_self">Th16</a></i>;<br/>
<i><font color="Green">E16</font></i>: <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> = 
 <a href="substut1.html#K39">CQC_Sub</a> <span class="p1"><a href="substut2.html#K2" target="_self">[</a><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>,<span class="p2">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span>

<br/>.= 
 <a href="cqc_lang.html#K10">'not'</a> <span class="p1">(<span class="default"><a href="substut1.html#K39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="substut2.html#E1:24"><i><font color="Green">E15</font></i></a>, <a class="ref" href="substut1.html#T29">SUBSTUT1:29</a></i>
;<br/>
<a NAME="E3:24"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K10">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
;<br/>
<a NAME="E4:24"/>
(  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> implies  <a href="cqc_sim1.html#K6">QuantNbr</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> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K6">QuantNbr</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="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_2"/><i><font color="Green">E18</font></i>: 
 <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:24_2"/>
 <a href="cqc_sim1.html#K6">QuantNbr</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="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="substut2.html#E3:24"><i><font color="Green">E17</font></i></a>, <a class="ref" href="cqc_sim1.html#T16">CQC_SIM1:16</a></i>;<br/>
<b>hence </b><a NAME="E3:24_2"/>
 <a href="cqc_sim1.html#K6">QuantNbr</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> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K6">QuantNbr</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="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="substut2.html#E1:24_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="cqc_sim1.html#T16">CQC_SIM1:16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:24"/>
(  <a href="cqc_lang.html#K10">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K10">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; (  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> implies  <a href="cqc_sim1.html#K6">QuantNbr</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> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="substut2.html#E2:24"><i><font color="Green">E16</font></i></a></i>;<br/>


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