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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <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">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:18"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="henmodel.html#V1" target="_self">Consistent</a>
 ;<br/>

<a NAME="E2:18"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1" target="_self">|-</a>  <a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a> 
 
<b>by </b><i><a class="ref" href="henmodel.html#T6" target="_self">Th6</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E4:18"/><i><font color="Green">E15</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="calcul_1.html#R4">|-</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="henmodel.html#D2" target="_self">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a>  as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  ;<br/>
<a NAME="E6:18"/>
(  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1">Ant</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="calcul_1.html#K2">Suc</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="henmodel.html#E4:18"><i><font color="Green">E15</font></i></a>, <a class="ref" href="calcul_1.html#T5">CALCUL_1:5</a></i>;<br/>
<a NAME="E7:18"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="calcul_1.html#R5">is_formal_provable_from</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="henmodel.html#E4:18"><i><font color="Green">E15</font></i></a>, <a class="ref" href="calcul_1.html#D10">CALCUL_1:def 10</a></i>;<br/>
<a NAME="E8:18"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="calcul_1.html#R7">|=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="calcul_1.html#T32">CALCUL_1:32</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="valuat_1.html#M1">interpretation</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">c<sub>2</sub></font>;<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">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="calcul_1.html#R6">|=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:18_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K10">'not'</a> <a href="cqc_lang.html#K9">VERUM</a> 
 <b>by </b><i><a class="txt" href="henmodel.html#E8:18"><i><font color="Green">E16</font></i></a>, <a class="ref" href="calcul_1.html#D12">CALCUL_1:def 12</a></i>;<br/><a NAME="E3:18_1"/><b>then </b>
not <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K9">VERUM</a> 
 <b>by </b><i><a class="ref" href="valuat_1.html#T28">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">VALUAT_1:44</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:18"/>
not <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="calcul_1.html#R6">|=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


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