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<div class="add">

<b>let </b><font color="Maroon" title="c1">p</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">x</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon" title="c4">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c5">v</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c4">A</font>;<br/><b>let </b><font color="Maroon" title="c6">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">A</font>;<br/>











<b>assume </b><a NAME="E1:44"/><i><font color="Green" title="E42">A1</font></i>: 
not <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c1">p</font></span>)</span>
 ;<br/>

<a NAME="E2:44"/>
<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#D1" title="SUBLEMMA:def.1">SUBLEMMA:def 1</a></i>;<br/>
<a NAME="E3:44"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#D1" title="SUBLEMMA:def.1">SUBLEMMA:def 1</a></i>;<br/>
<a NAME="E4:44"/><b>then </b><i><font color="Green" title="E43">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T15" title="FUNCT_4:th.15">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" title="E44">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font>
 ;<br/><a NAME="E2:44_1"/>
not <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:44"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="qc_lang3.html#T16" title="QC_LANG3:th.16">QC_LANG3:16</a></i>;<br/><a NAME="E3:44_1"/><b>then </b><i><font color="Green" title="E45">A4</font></i>: 
not <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E1:44_1"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><a NAME="E4:44_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="sublemma.html#T59" title="SUBLEMMA:th.59">SUBLEMMA:59</a></i>;<br/><a NAME="E5:44_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_1.html#E4:44"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="calcul_1.html#E1:44_1"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="funct_4.html#T36" title="FUNCT_4:th.36">FUNCT_4:36</a>, <a class="ref" href="zfmisc_1.html#T17" title="ZFMISC_1:th.17">ZFMISC_1:17</a></i>;<br/><a NAME="E6:44_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T15" title="FUNCT_4:th.15">FUNCT_4:15</a></i>;<br/><a NAME="E7:44_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="sublemma.html#D1" title="SUBLEMMA:def.1">SUBLEMMA:def 1</a></i>;<br/><a NAME="E8:44_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="sublemma.html#D1" title="SUBLEMMA:def.1">SUBLEMMA:def 1</a></i>;<br/><b>hence </b><a NAME="E9:44_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:44_1"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="calcul_1.html#T26" target="_self" title="CALCUL_1:th.26">Th26</a></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" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="calcul_1.html#E4:44"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="sublemma.html#D1" title="SUBLEMMA:def.1">SUBLEMMA:def 1</a></i>;<br/>


</div>
