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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</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>6</sub></font> be    <a href="valuat_1.html#M1" target="_self">interpretation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="qc_lang1.html#NM8">QC-formula</a>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:63"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>and </b><br/><a NAME="E2:63"/><i><font color="Green">E41</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E3:63"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="valuat_1.html#R2" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:63_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:63_1_1"/><i><font color="Green">E45</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>from </b><i><a class="ref" href="valuat_1.html#S1" target="_self">VALUAT_1:sch 1</a>();<br/></i><a NAME="E3:63_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:63_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E4:63_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:63"><i><font color="Green">E40</font></i></a>, <a class="ref" href="valuat_1.html#T42" target="_self">Th42</a></i>;<br/><a NAME="E5:63_1_1"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="valuat_1.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E3:63"><i><font color="Green">E42</font></i></a>, <a class="ref" href="valuat_1.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E6:63_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="valuat_1.html#D12" target="_self">Def12</a></i>;<br/><a NAME="E7:63_1_1"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:63"><i><font color="Green">E40</font></i></a>, <a class="txt" href="valuat_1.html#E2:63"><i><font color="Green">E41</font></i></a>, <a class="txt" href="valuat_1.html#E1:63_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="valuat_1.html#T43" target="_self">Th43</a></i>;<br/><b>hence </b><a NAME="E8:63_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:63_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="valuat_1.html#E4:63_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="valuat_1.html#E6:63_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="valuat_1.html#T39" target="_self">Th39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:63_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:63_1"><i><font color="Green">E44</font></i></a></i>;<br/><b>hence </b><a NAME="E2:63_1_2"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="valuat_1.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="valuat_1.html#D12" target="_self">Def12</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:63_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="valuat_1.html#R2" target="_self">|=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="valuat_1.html#D13" target="_self">Def13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:63"/>
<font color="Maroon">c<sub>6</sub></font> <a href="valuat_1.html#R2" target="_self">|=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:63"><i><font color="Green">E40</font></i></a>, <a class="txt" href="valuat_1.html#E3:63"><i><font color="Green">E42</font></i></a></i>;<br/>


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