<?xml version="1.0"?>
<div class="add">

<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  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>4</sub></font> be    <a href="valuat_1.html#M1">interpretation</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>







<b>assume </b><a NAME="E1:84"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>1</sub></font> <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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <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">c<sub>1</sub></font></span>)</span> holds <br/>( <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>1</sub></font> iff <font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</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>3</sub></font>;<br/>



<b>assume </b><a NAME="E2:84"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>;<br/>
<a NAME="E3:84"/>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T16">QC_LANG3:16</a></i>;<br/>
<a NAME="E4:84"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E2:84"><i><font color="Green">E67</font></i></a>, <a class="ref" href="qc_lang3.html#T16">QC_LANG3:16</a></i>;<br/>
<a NAME="E5:84"/><i><font color="Green">E69</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</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> ) iff for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</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="84_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:84_1"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</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> ) implies for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</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="84_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_1_1"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</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>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<a NAME="E2:84_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E4:84"><i><font color="Green">E68</font></i></a>, <a class="ref" href="sublemma.html#T67" target="_self">Th67</a></i>;<br/>
<a NAME="E3:84_1_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</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> iff <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</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="sublemma.html#E1:84"><i><font color="Green">E66</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:84_1_1"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</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="sublemma.html#E1:84_1_1"><i><font color="Green">E70</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:84_1"/>
( ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</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> ) implies for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</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="84_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_1_2"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</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>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<a NAME="E2:84_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</a> <font color="Maroon">c<sub>8</sub></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">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E4:84"><i><font color="Green">E68</font></i></a>, <a class="ref" href="sublemma.html#T67" target="_self">Th67</a></i>;<br/>
<a NAME="E3:84_1_2"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</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> iff <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</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="sublemma.html#E1:84"><i><font color="Green">E66</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:84_1_2"/>
<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="sublemma.html#K13" target="_self">|</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="sublemma.html#E1:84_1_2"><i><font color="Green">E70</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>thus </b><a NAME="E6:84"/>
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</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> iff <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</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="sublemma.html#E5:84"><i><font color="Green">E69</font></i></a>, <a class="ref" href="sublemma.html#T51" target="_self">Th51</a></i>;<br/>


</div>
