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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>









<span class="kw">thus </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><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/>
<span class="kw">reconsider </span><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/>
<span class="kw">assume </span><a NAME="E2:28_1"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<a NAME="E3:28_1"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valuat_1.html#D12" title="VALUAT_1:def.12">VALUAT_1:def 12</a></span>;<br/>
<a NAME="E4:28_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valuat_1.html#T16" title="VALUAT_1:th.16">VALUAT_1:16</a></span>;<br/>
<a NAME="E5:28_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="henmodel.html#T14" target="_self" title="HENMODEL:th.14">Th14</a></span>;<br/>
<a NAME="E6:28_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">ll9</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> st <br/>( <font color="Olive" title="b1">ll9</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="Olive" title="b1">ll9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="henmodel.html#D6" target="_self" title="HENMODEL:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:28_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:28_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T3" title="QC_LANG3:th.3">QC_LANG3:3</a></span>;<br/>
<a NAME="E2:28_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substut2.html#T3" title="SUBSTUT2:th.3">SUBSTUT2:3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:28_2"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E4:28_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_2"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="henmodel.html#D6" target="_self" title="HENMODEL:def.6">Def6</a></span>;<br/>
<a NAME="E5:28_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="henmodel.html#T14" target="_self" title="HENMODEL:th.14">Th14</a></span>;<br/>
<a NAME="E6:28_2"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valuat_1.html#T16" title="VALUAT_1:th.16">VALUAT_1:16</a></span>;<br/>
<span class="kw">hence </span><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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valuat_1.html#D12" title="VALUAT_1:def.12">VALUAT_1:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
