<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>

<b>thus </b><a NAME="E1:23"/>
(  <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>1</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a>  implies <font color="Maroon">c<sub>1</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1"/>
 <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>1</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a> 
 ;<br/>

<a NAME="E2:23_1"/><b>then </b>
(  <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="qc_lang3.html#K5">Free</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E3:23_1"/><b>then </b>
(  <a href="qc_lang3.html#K6">Fixed</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="qc_lang3.html#K5">Free</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T67">QC_LANG3:67</a>, <a class="ref" href="qc_lang3.html#T78">QC_LANG3:78</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1"/>
<font color="Maroon">c<sub>1</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a> 
 <b>by </b><i><a class="ref" href="cqc_lang.html#T13" target="_self">Th13</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:23"/>
<font color="Maroon">c<sub>1</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a>  ;<br/>
<a NAME="E4:23"/>
(  <a href="qc_lang3.html#K5">Free</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="qc_lang3.html#K6">Fixed</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E5:23"/><b>then </b>
(  <a href="qc_lang3.html#K5">Free</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="qc_lang3.html#K6">Fixed</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T67">QC_LANG3:67</a>, <a class="ref" href="qc_lang3.html#T78">QC_LANG3:78</a></i>;<br/>
<b>hence </b><a NAME="E6:23"/>
 <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>1</sub></font> is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a> 
 <b>by </b><i><a class="ref" href="cqc_lang.html#T13" target="_self">Th13</a></i>;<br/>


</div>
