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<b>let </b><font color="Maroon">p</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">y</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><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">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">v</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>











<b>assume </b><a NAME="E1:44"/><i><font color="Green">E52</font></i>: 
not <font color="Maroon">y</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">x</font>,<font color="Maroon">p</font></span>)</span>
 ;<br/>

<a NAME="E2:44"/>
<font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="sublemma.html#K15">+*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#D1">SUBLEMMA:def 1</a></i>;<br/>
<a NAME="E3:44"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K15">+*</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sublemma.html#K15">+*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#D1">SUBLEMMA:def 1</a></i>;<br/>
<a NAME="E4:44"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="sublemma.html#K15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T15">FUNCT_4:15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:44_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:44_1"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="qc_lang3.html#T16">QC_LANG3:16</a></i>;<br/><a NAME="E3:44_1"/><b>then </b><i><font color="Green">E55</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/><a NAME="E4:44_1"/>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="sublemma.html#T59">SUBLEMMA:59</a></i>;<br/><a NAME="E5:44_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T17">ZFMISC_1:17</a></i>;<br/><a NAME="E6:44_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="sublemma.html#K15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span> <a href="sublemma.html#K15">+*</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_4.html#T36">FUNCT_4:36</a></i>;<br/><a NAME="E7:44_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K15">+*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sublemma.html#K15">+*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T15">FUNCT_4:15</a></i>;<br/><a NAME="E8:44_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sublemma.html#K15">+*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="sublemma.html#D1">SUBLEMMA:def 1</a></i>;<br/><a NAME="E9:44_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="sublemma.html#D1">SUBLEMMA:def 1</a></i>;<br/><b>hence </b><a NAME="E10:44_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:44"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="sublemma.html#K13">|</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="sublemma.html#D1">SUBLEMMA:def 1</a></i>;<br/>


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