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<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  <a href="henmodel.html#V1">Consistent</a> <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="henmodel.html#M1">Henkin_interpretation</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>







<b>thus </b><a NAME="E1:9"/>
( <font color="Maroon">c<sub>4</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st <font color="Maroon">c<sub>4</sub></font>, <a href="henmodel.html#K4">valH</a>  <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="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1"/>
<font color="Maroon">c<sub>4</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="henmodel.html#K2">HCar</a> <b> such that </b><br/><a NAME="E3:9_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>4</sub></font>,<a href="henmodel.html#K4">valH</a>  <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>5</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="ref" href="goedelcp.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E4:9_1"/>
(  <a href="relset_1.html#K5">rng</a> <a href="henmodel.html#K4">valH</a>  <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  &amp;  <a href="relset_1.html#K4">dom</a> <a href="henmodel.html#K4">valH</a>  <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  )
 
<b>by </b><i><a class="ref" href="henmodel.html#D7">HENMODEL:def 7</a>, <a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:9_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <a href="henmodel.html#K4">valH</a>  &amp; <a href="henmodel.html#K4">valH</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="henmodel.html#D5">HENMODEL:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="qc_lang1.html#NM3">bound_QC-variable</a> <b>by </b><i><a class="txt" href="goedelcp.html#E6:9_1"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E8:9_1"/>
<font color="Maroon">c<sub>4</sub></font>, <a href="henmodel.html#K4">valH</a>  <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>7</sub></font>
 <b>by </b><i><a class="txt" href="goedelcp.html#E3:9_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="goedelcp.html#E6:9_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="calcul_1.html#T24">CALCUL_1:24</a></i>;<br/>


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

<b>thus </b><a NAME="E2:9"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st <font color="Maroon">c<sub>4</sub></font>, <a href="henmodel.html#K4">valH</a>  <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="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>4</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>5</sub></font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><a NAME="E1:9_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>4</sub></font>, <a href="henmodel.html#K4">valH</a>  <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>5</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="henmodel.html#K2">HCar</a> <b> such that </b><br/><a NAME="E3:9_2"/><i><font color="Green">E14</font></i>: 
( <a href="henmodel.html#K4">valH</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<a href="henmodel.html#K4">valH</a>  <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>6</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="goedelcp.html#E1:9_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="calcul_1.html#T24">CALCUL_1:24</a></i>;<br/>
<b>thus </b><a NAME="E4:9_2"/>
<font color="Maroon">c<sub>4</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K16">Ex</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="goedelcp.html#E3:9_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="goedelcp.html#T9" target="_self">Th9</a></i>;<br/>


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


</div>
