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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">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</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">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>







<b>assume </b><a NAME="E1:45"/><i><font color="Green">E52</font></i>: 
(  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span> &amp; 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/>

<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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="calcul_1.html#D15" target="_self">CALCUL_1:def 15</a><br/><b>let </b><font color="Maroon">J</font> be    <a href="valuat_1.html#M1">interpretation</a> of <font color="Maroon">A</font>;<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>assume </b><a NAME="E2:45"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="calcul_1.html#R9" target="_self">|=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E3:45"/>
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> holds  <font color="Maroon">J</font>,<font color="Maroon">v</font> <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="Olive">a</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E1:45_1"/>
<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="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" target="_self">still_not-bound_in</a> <span class="p2">(<span class="default"><a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:45_1"/><b>then </b>
<font color="Maroon">J</font>,<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="calcul_1.html#R9" target="_self">|=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:45_1"/><b>then </b>
<font color="Maroon">J</font>,<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="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">a1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E5:45_1"/><i><font color="Green">E54</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="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">J</font>,<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">a1</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:45_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">J</font>,<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="valuat_1.html#R1">|=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="sublemma.html#T50">SUBLEMMA:50</a></i>;<br/>
<a NAME="E7:45_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/>
<b>hence </b><a NAME="E8:45_1"/>
<font color="Maroon">J</font>,<font color="Maroon">v</font> <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="valuat_1.html#R1">|=</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="sublemma.html#T70">SUBLEMMA:70</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:45"/>
<font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="sublemma.html#T51">SUBLEMMA:51</a></i>;<br/>


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