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<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">ll</font> be   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">CX</font> be  <a href="henmodel.html#V1" target="_self">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">JH</font> be    <a href="henmodel.html#M1" target="_self">Henkin_interpretation</a> of <font color="Maroon">CX</font>;<br/>









<b>thus </b><a NAME="E1:28"/>
( <font color="Maroon">JH</font>, <a href="henmodel.html#K4" target="_self">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font> implies <font color="Maroon">CX</font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">JH</font>, <a href="henmodel.html#K4" target="_self">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font>
 ;<br/>

<b>set </b><font color="Maroon">rel</font> = <font color="Maroon">JH</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">P</font>;<br/>
<b>reconsider </b><font color="Maroon">rel</font> = <font color="Maroon">JH</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">P</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K3">relations_on</a> <a href="henmodel.html#K2" target="_self">HCar</a>  ;<br/>
<a NAME="E3:28_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font></span>)</span>,<font color="Maroon">JH</font></span>)</span> <a href="funct_2.html#K3">.</a> <a href="henmodel.html#K4" target="_self">valH</a>  <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">VALUAT_1:def 12</a></i>;<br/>
<a NAME="E4:28_1"/><b>then </b>
<a href="henmodel.html#K4" target="_self">valH</a>  <a href="valuat_1.html#K8">*'</a> <font color="Maroon">ll</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rel</font>
 
<b>by </b><i><a class="ref" href="valuat_1.html#T16">VALUAT_1:16</a></i>;<br/>
<a NAME="E5:28_1"/><b>then </b>
<font color="Maroon">ll</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rel</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">ll'</font> being   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">P</font><b> such that </b><br/><a NAME="E7:28_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">ll'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ll</font> &amp; <font color="Maroon">CX</font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">P</font> <a href="henmodel.html#K3" target="_self">!</a> <font color="Maroon">ll'</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E8:28_1"/>
<font color="Maroon">CX</font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:28"/>
( <font color="Maroon">CX</font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font> implies <font color="Maroon">JH</font>, <a href="henmodel.html#K4" target="_self">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">CX</font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font>
 ;<br/>

<a NAME="E2:28_2"/>
<font color="Maroon">P</font> is   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T3">QC_LANG3:3</a></i>;<br/>
<a NAME="E3:28_2"/><b>then </b>
 <a href="qc_lang1.html#K6">the_arity_of</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="substut2.html#T3">SUBSTUT2:3</a></i>;<br/>
<a NAME="E4:28_2"/><b>then </b>
<font color="Maroon">ll</font> <a href="hidden.html#R2">in</a> <font color="Maroon">JH</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">P</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:28_2"/><b>then </b>
<a href="henmodel.html#K4" target="_self">valH</a>  <a href="valuat_1.html#K8">*'</a> <font color="Maroon">ll</font> <a href="hidden.html#R2">in</a> <font color="Maroon">JH</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">P</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:28_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font></span>)</span>,<font color="Maroon">JH</font></span>)</span> <a href="funct_2.html#K3">.</a> <a href="henmodel.html#K4" target="_self">valH</a>  <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>hence </b><a NAME="E7:28_2"/>
<font color="Maroon">JH</font>, <a href="henmodel.html#K4" target="_self">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">ll</font>
 <b>by </b><i><a class="ref" href="valuat_1.html#D12">VALUAT_1:def 12</a></i>;<br/>


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


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