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<span class="kw">let </span><font color="Maroon" title="c1">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#K3" title="CQC_LANG:func.3">CQC-WFF</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c1">CX</font><br/>  for <font color="Olive" title="b2">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">CX</font> is  <a href="goedelcp.html#V1" title="GOEDELCP:attr.1">negation_faithful</a>  &amp; <font color="Maroon" title="c1">CX</font> is  <a href="goedelcp.html#V2" title="GOEDELCP:attr.2">with_examples</a>  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="Olive" title="b2">p</font> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b2">p</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">JH</font> be    <a href="henmodel.html#M1" title="HENMODEL:mode.1">Henkin_interpretation</a> of <font color="Maroon" title="c1">CX</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">CX</font> is  <a href="goedelcp.html#V1" title="GOEDELCP:attr.1">negation_faithful</a>  &amp; <font color="Maroon" title="c1">CX</font> is  <a href="goedelcp.html#V2" title="GOEDELCP:attr.2">with_examples</a>  holds <br/>( <font color="Maroon" title="c2">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> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p</font> )</span><br/>



<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> ] <span class="kw">means </span>( <font color="Maroon" title="c1">CX</font> is  <a href="goedelcp.html#V1" title="GOEDELCP:attr.1">negation_faithful</a>  &amp; <font color="Maroon" title="c1">CX</font> is  <a href="goedelcp.html#V2" title="GOEDELCP:attr.2">with_examples</a>  implies ( <font color="Maroon" title="c2">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> $1 iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> $1 ) );<br/>
<a NAME="E1:7"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
 for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <br/>  for <font color="Olive" title="b3">x</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/>  for <font color="Olive" title="b4">k</font> being    <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> <br/>  for <font color="Olive" title="b5">l</font> being   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Olive" title="b4">k</font><br/>  for <font color="Olive" title="b6">P</font> being   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Olive" title="b4">k</font> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a> ] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b6">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Olive" title="b5">l</font>] &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Olive" title="b1">r</font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Olive" title="b2">s</font>] ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goedelcp.html#D1" target="_self" title="GOEDELCP:def.1">Def1</a>, <a class="ref" href="goedelcp.html#T6" target="_self" title="GOEDELCP:th.6">Th6</a>, <a class="ref" href="henmodel.html#T16" title="HENMODEL:th.16">HENMODEL:16</a>, <a class="ref" href="henmodel.html#T17" title="HENMODEL:th.17">HENMODEL:17</a>, <a class="ref" href="henmodel.html#D2" title="HENMODEL:def.2">HENMODEL:def 2</a>, <a class="ref" href="valuat_1.html#T17" title="VALUAT_1:th.17">VALUAT_1:17</a>, <a class="ref" href="valuat_1.html#T18" title="VALUAT_1:th.18">VALUAT_1:18</a></span>;<br/>
<a NAME="E2:7"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="substut2.html#S3" title="SUBSTUT2:sch.3">SUBSTUT2:sch 3</a>(<a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E8">A1</font></span></a>);<br/></span>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">p</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:7_1"/>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font>]</span><br/><a NAME="E2:7_1"/><span class="kw">then </span>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:7_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E9">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:7"/>
 for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">CX</font> is  <a href="goedelcp.html#V1" title="GOEDELCP:attr.1">negation_faithful</a>  &amp; <font color="Maroon" title="c1">CX</font> is  <a href="goedelcp.html#V2" title="GOEDELCP:attr.2">with_examples</a>  holds <br/>( <font color="Maroon" title="c2">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> iff <font color="Maroon" title="c1">CX</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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