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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</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="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38">CQC-Sub-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="substut1.html#M1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>;<br/>









<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font>;<br/>
<b>assume </b><a NAME="E1:106"/><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span> is <a href="substut1.html#V3">quantifiable</a>
 ;<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>2</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="sublemma.html#NM1">Val_Sub</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:106"/><i><font color="Green">E90</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"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span></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>7</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> <span class="p1">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <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> )
 <b>by </b><i><a class="txt" href="sublemma.html#E1:106"><i><font color="Green">E89</font></i></a>, <a class="ref" href="sublemma.html#T90" target="_self">Th90</a></i>;<br/>
<a NAME="E4:106"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span> iff <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <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>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="sublemma.html#E3:106"><i><font color="Green">E90</font></i></a>, <a class="ref" href="sublemma.html#T85" target="_self">Th85</a></i>;<br/>
<a NAME="E5:106"/>
 <a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K24">Sub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:106"><i><font color="Green">E89</font></i></a>, <a class="ref" href="sublemma.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E6:106"/><b>then </b>
<span class="p1">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a>  <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K15">All</a> <span class="p1">(<span class="default"><span class="p2"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span> <a href="substut1.html#K23">`2</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span> <a href="sublemma.html#K9" target="_self">`1</a> </span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:106"><i><font color="Green">E89</font></i></a>, <a class="ref" href="sublemma.html#T27" target="_self">Th27</a></i>;<br/>
<a NAME="E7:106"/><b>then </b>
<span class="p1">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a>  <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span> <a href="sublemma.html#K9" target="_self">`1</a> </span>)</span> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E8:106"/><b>then </b>
<span class="p1">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p2"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a>  <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E9:106"/>
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K14" target="_self">NEx_Val</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span> iff <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p3"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a>  <a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="sublemma.html#E3:106"><i><font color="Green">E90</font></i></a>, <a class="txt" href="sublemma.html#E4:106"><i><font color="Green">E91</font></i></a>, <a class="ref" href="sublemma.html#D3" target="_self">Def3</a></i>;<br/>


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