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<div class="add">

<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">p</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="E1:30"/>
(  <a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</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">p</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="30_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1"/>
 <a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</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="E2:30_1"/><b>then </b>
(  <a href="qc_lang3.html#K5">Free</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</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#K14">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E3:30_1"/><b>then </b>
(  <a href="qc_lang3.html#K5">Free</a> <font color="Maroon">p</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">p</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#T70">QC_LANG3:70</a>, <a class="ref" href="qc_lang3.html#T81">QC_LANG3:81</a></i>;<br/>
<b>hence </b><a NAME="E4:30_1"/>
<font color="Maroon">p</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/>;<br/>


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

<b>assume </b><a NAME="E2:30"/>
<font color="Maroon">p</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="E3:30"/><b>then </b>
(  <a href="qc_lang3.html#K5">Free</a> <font color="Maroon">p</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">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E4:30"/><b>then </b>
(  <a href="qc_lang3.html#K5">Free</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</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#K14">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</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#T70">QC_LANG3:70</a>, <a class="ref" href="qc_lang3.html#T81">QC_LANG3:81</a></i>;<br/>
<b>hence </b><a NAME="E5:30"/>
 <a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</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/>;<br/>


</div>
