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<b>let </b><font color="Maroon" title="c1">X</font> be   <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">A</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st not <font color="Maroon" title="c1">X</font> is <a href="henmodel.html#V1" title="HENMODEL:attr.1">Consistent</a> holds <br/>for <font color="Olive" title="b2">J</font> being   <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Olive" title="b1">A</font><br/> for <font color="Olive" title="b3">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Olive" title="b1">A</font>  holds not <font color="Olive" title="b2">J</font>,<font color="Olive" title="b3">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a> <font color="Maroon" title="c1">X</font></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c1">X</font> is <a href="henmodel.html#V1" title="HENMODEL:attr.1">Consistent</a> implies for <font color="Olive" title="b1">J</font> being   <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font>  holds not <font color="Olive" title="b1">J</font>,<font color="Olive" title="b2">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a> <font color="Maroon" title="c1">X</font> )</span><br/>



<b>assume </b><a NAME="E1:18"/>
not <font color="Maroon" title="c1">X</font> is <a href="henmodel.html#V1" title="HENMODEL:attr.1">Consistent</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">J</font> being   <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font>  holds not <font color="Olive" title="b1">J</font>,<font color="Olive" title="b2">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:18"/><b>then </b>
<font color="Maroon" title="c1">X</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a>  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> 
 
<b>by </b><i><a class="ref" href="henmodel.html#T6" target="_self" title="HENMODEL:th.6">Th6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E4:18"/><i><font color="Green" title="E13">A1</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="calcul_1.html#R4" title="CALCUL_1:pred.4">|-</a> <font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="henmodel.html#D2" target="_self" title="HENMODEL:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">p</font> =  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  ;<br/>
<a NAME="E6:18"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> )
 
<b>by </b><i><a class="txt" href="#E4:18"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="calcul_1.html#T5" title="CALCUL_1:th.5">CALCUL_1:5</a></i>;<br/>
<a NAME="E7:18"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="calcul_1.html#R5" title="CALCUL_1:pred.5">is_formal_provable_from</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="#E4:18"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="calcul_1.html#D10" title="CALCUL_1:def.10">CALCUL_1:def 10</a></i>;<br/>
<a NAME="E8:18"/><b>then </b><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="calcul_1.html#R7" title="CALCUL_1:pred.7">|=</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="ref" href="calcul_1.html#T32" title="CALCUL_1:th.32">CALCUL_1:32</a></i>;<br/>
<b>let </b><font color="Maroon" title="c5">J</font> be    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c2">A</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">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font>  holds not <font color="Maroon" title="c5">J</font>,<font color="Olive" title="b1">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a> <font color="Maroon" title="c1">X</font></span><br/><b>let </b><font color="Maroon" title="c6">v</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c6">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a> <font color="Maroon" title="c1">X</font></span><br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c6">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:18_1"/><b>then </b>
<font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c6">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> 
 <b>by </b><i><a class="txt" href="#E8:18"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="calcul_1.html#D12" title="CALCUL_1:def.12">CALCUL_1:def 12</a></i>;<br/><a NAME="E3:18_1"/><b>then </b>
not <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c6">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> 
 <b>by </b><i><a class="ref" href="valuat_1.html#T28" title="VALUAT_1:th.28">VALUAT_1:28</a></i>;<br/><b>hence </b><a NAME="E4:18_1"/>
contradiction
 <b>by </b><i><a class="ref" href="valuat_1.html#T44" title="VALUAT_1:th.44">VALUAT_1:44</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>hence </b><a NAME="E10:18"/>
not <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c6">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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