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

<b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b2">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b3">CX</font> being <a href="henmodel.html#V1" title="HENMODEL:attr.1">Consistent</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <br/> for <font color="Olive" title="b4">JH</font> being   <a href="henmodel.html#M1" title="HENMODEL:mode.1">Henkin_interpretation</a> of <font color="Olive" title="b3">CX</font> holds <br/> ( <font color="Olive" title="b4">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b2">ll</font> iff <font color="Olive" title="b3">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b2">ll</font> )</span><br/><b>let </b><font color="Maroon" title="c2">P</font> be   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b2">CX</font> being <a href="henmodel.html#V1" title="HENMODEL:attr.1">Consistent</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <br/> for <font color="Olive" title="b3">JH</font> being   <a href="henmodel.html#M1" title="HENMODEL:mode.1">Henkin_interpretation</a> of <font color="Olive" title="b2">CX</font> holds <br/> ( <font color="Olive" title="b3">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b1">ll</font> iff <font color="Olive" title="b2">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b1">ll</font> )</span><br/><b>let </b><font color="Maroon" title="c3">ll</font> be   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">CX</font> being <a href="henmodel.html#V1" title="HENMODEL:attr.1">Consistent</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <br/> for <font color="Olive" title="b2">JH</font> being   <a href="henmodel.html#M1" title="HENMODEL:mode.1">Henkin_interpretation</a> of <font color="Olive" title="b1">CX</font> holds <br/> ( <font color="Olive" title="b2">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font> iff <font color="Olive" title="b1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font> )</span><br/><b>let </b><font color="Maroon" title="c4">CX</font> be  <a href="henmodel.html#V1" title="HENMODEL:attr.1">Consistent</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">JH</font> being   <a href="henmodel.html#M1" title="HENMODEL:mode.1">Henkin_interpretation</a> of <font color="Maroon" title="c4">CX</font> holds <br/> ( <font color="Olive" title="b1">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font> iff <font color="Maroon" title="c4">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font> )</span><br/><b>let </b><font color="Maroon" title="c5">JH</font> be    <a href="henmodel.html#M1" title="HENMODEL:mode.1">Henkin_interpretation</a> of <font color="Maroon" title="c4">CX</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font> iff <font color="Maroon" title="c4">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font> )</span><br/>









<b>thus </b><a NAME="E1:28"/>
( <font color="Maroon" title="c5">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font> implies <font color="Maroon" title="c4">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font> implies <font color="Maroon" title="c5">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font> )</span><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" title="E17">A1</font></i>: 
<font color="Maroon" title="c5">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font></span><br/>

<b>set </b><font color="Maroon" title="c6">rel</font> = <font color="Maroon" title="c5">JH</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">P</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">rel</font> = <font color="Maroon" title="c5">JH</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">P</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K3" title="MARGREL1:func.3">relations_on</a> <a href="henmodel.html#K1" title="HENMODEL:func.1">HCar</a>  ;<br/>
<a NAME="E3:28_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font></span>)</span>,<font color="Maroon" title="c5">JH</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="henmodel.html#E1:28_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="valuat_1.html#D12" title="VALUAT_1:def.12">VALUAT_1:def 12</a></i>;<br/>
<a NAME="E4:28_1"/><b>then </b>
<a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c3">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">rel</font>
 
<b>by </b><i><a class="ref" href="valuat_1.html#T16" title="VALUAT_1:th.16">VALUAT_1:16</a></i>;<br/>
<a NAME="E5:28_1"/><b>then </b>
<font color="Maroon" title="c3">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">rel</font>
 
<b>by </b><i><a class="ref" href="henmodel.html#T14" target="_self" title="HENMODEL:th.14">Th14</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">ll'</font> being   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of  <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c2">P</font><b> such that </b><br/><a NAME="E7:28_1"/><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c8">ll'</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">ll</font> &amp; <font color="Maroon" title="c4">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c2">P</font> <a href="henmodel.html#K2" title="HENMODEL:func.2">!</a> <font color="Maroon" title="c8">ll'</font> )
 <b>by </b><i><a class="ref" href="henmodel.html#D6" target="_self" title="HENMODEL:def.6">Def6</a></i>;<br/>
<b>thus </b><a NAME="E8:28_1"/>
<font color="Maroon" title="c4">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font>
 <b>by </b><i><a class="txt" href="henmodel.html#E7:28_1"><i><font color="Green" title="E18">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:28"/>
( <font color="Maroon" title="c4">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font> implies <font color="Maroon" title="c5">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E17">A3</font></i>: 
<font color="Maroon" title="c4">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font></span><br/>

<a NAME="E2:28_2"/>
<font color="Maroon" title="c2">P</font> is   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c2">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T3" title="QC_LANG3:th.3">QC_LANG3:3</a></i>;<br/>
<a NAME="E3:28_2"/><b>then </b>
 <a href="qc_lang1.html#K6" title="QC_LANG1:func.6">the_arity_of</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="ref" href="substut2.html#T3" title="SUBSTUT2:th.3">SUBSTUT2:3</a></i>;<br/>
<a NAME="E4:28_2"/><b>then </b>
<font color="Maroon" title="c3">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">JH</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">P</font>
 
<b>by </b><i><a class="txt" href="henmodel.html#E1:28_2"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="henmodel.html#D6" target="_self" title="HENMODEL:def.6">Def6</a></i>;<br/>
<a NAME="E5:28_2"/><b>then </b>
<a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c3">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">JH</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">P</font>
 
<b>by </b><i><a class="ref" href="henmodel.html#T14" target="_self" title="HENMODEL:th.14">Th14</a></i>;<br/>
<a NAME="E6:28_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font></span>)</span>,<font color="Maroon" title="c5">JH</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="valuat_1.html#T16" title="VALUAT_1:th.16">VALUAT_1:16</a></i>;<br/>
<b>hence </b><a NAME="E7:28_2"/>
<font color="Maroon" title="c5">JH</font>, <a href="henmodel.html#K3" title="HENMODEL:func.3">valH</a>  <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c3">ll</font>
 <b>by </b><i><a class="ref" href="valuat_1.html#D12" title="VALUAT_1:def.12">VALUAT_1:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div>
