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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</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>3</sub></font> be    <a href="valuat_1.html#M1">interpretation</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><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>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="sublemma.html#NM1">Val_Sub</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

















<b>assume </b><a NAME="E1:92"/><i><font color="Green">E75</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#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> holds <br/> not <font color="Olive">b<sub>1</sub></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"><font color="Maroon">c<sub>4</sub></font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> ) &amp; ( 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#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/>

<a NAME="E2:92"/><i><font color="Green">E76</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#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> holds <br/> not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/>

<b>assume </b><a NAME="E1:92_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:92_1"/><b>then </b>
not <font color="Maroon">c<sub>10</sub></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"><font color="Maroon">c<sub>4</sub></font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:92"><i><font color="Green">E75</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:92_1"/>
not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K22">still_not-bound_in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="qc_lang3.html#T9">QC_LANG3:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:92"/><i><font color="Green">E77</font></i>: 
( <span class="p1">(<span class="default"><a href="valuat_1.html#K12">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  iff <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="valuat_1.html#K8">*'</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="valuat_1.html#T16">VALUAT_1:16</a></i>;<br/>
<a NAME="E4:92"/><i><font color="Green">E78</font></i>: 
( <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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="valuat_1.html#K8">*'</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> iff <span class="p1">(<span class="default"><a href="valuat_1.html#K12">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 
<b>by </b><i><a class="ref" href="valuat_1.html#T16">VALUAT_1:16</a></i>;<br/>
<b>thus </b><a NAME="E5:92"/>
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">c<sub>5</sub></font> iff <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="sublemma.html#E1:92"><i><font color="Green">E75</font></i></a>, <a class="txt" href="sublemma.html#E2:92"><i><font color="Green">E76</font></i></a>, <a class="txt" href="sublemma.html#E3:92"><i><font color="Green">E77</font></i></a>, <a class="txt" href="sublemma.html#E4:92"><i><font color="Green">E78</font></i></a>, <a class="ref" href="sublemma.html#T78" target="_self">Th78</a>, <a class="ref" href="valuat_1.html#D12">VALUAT_1:def 12</a></i>;<br/>


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