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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></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="E1:41"/><i><font color="Green">E39</font></i>: 
(  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">c<sub>3</sub></font> <a href="calcul_1.html#R10" target="_self">|=</a>  <a href="calcul_1.html#K2" target="_self">Suc</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></font> be    <a href="valuat_1.html#M1">interpretation</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">c<sub>5</sub></font>;<br/>







<b>assume </b><a NAME="E2:41"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="calcul_1.html#R9" target="_self">|=</a>  <a href="calcul_1.html#K1" target="_self">Ant</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:41"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="calcul_1.html#E1:41"><i><font color="Green">E39</font></i></a>, <a class="txt" href="calcul_1.html#E2:41"><i><font color="Green">E40</font></i></a>, <a class="ref" href="calcul_1.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E4:41"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/>

<b>thus </b><a NAME="E1:41_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="sublemma.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="calcul_1.html#E3:41"><i><font color="Green">E41</font></i></a>, <a class="ref" href="sublemma.html#T51">SUBLEMMA:51</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:41"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>1</sub></font> <a href="substut2.html#K4">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="calcul_1.html#T24" target="_self">Th24</a></i>;<br/>


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