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<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>

<b>thus </b><a NAME="E1:12"/>
( not <font color="Maroon">X</font> is <a href="henmodel.html#V1" target="_self">Consistent</a> implies for <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  holds  <font color="Maroon">X</font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Olive">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/>
not <font color="Maroon">X</font> is <a href="henmodel.html#V1" target="_self">Consistent</a>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:12_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">X</font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Maroon">q</font> &amp; <font color="Maroon">X</font> <a href="henmodel.html#R1" target="_self">|-</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q</font> )
 <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>

<b>consider </b><font color="Maroon">f1</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="E5:12_1"/><i><font color="Green">E48</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="calcul_1.html#R4">|-</a> <font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">q</font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">f2</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="E7:12_1"/><i><font color="Green">E49</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="calcul_1.html#R4">|-</a> <font color="Maroon">f2</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> <font color="Maroon">q</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:12_1"/><i><font color="Green">E50</font></i>: 
(  <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f2</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">q</font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp;  <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f2</font></span>)</span> <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> <font color="Maroon">q</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 
<b>by </b><i>, , , <a class="ref" href="calcul_2.html#T20">CALCUL_2:20</a></i>;<br/>
<b>take </b><font color="Maroon">f3</font> = <font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="henmodel.html#D2" target="_self">HENMODEL:def 2</a><br/>

<a NAME="E9:12_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<b>hence </b><a NAME="E10:12_1"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="calcul_1.html#R4">|-</a> <font color="Maroon">f3</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, , , <a class="ref" href="calcul_2.html#T25">CALCUL_2:25</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>


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

<b>assume </b><a NAME="E2:12"/>
for <font color="Olive">p</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  holds  <font color="Maroon">X</font> <a href="henmodel.html#R1" target="_self">|-</a> <font color="Olive">p</font>
 ;<br/>

<a NAME="E3:12"/><b>then </b>
( <font color="Maroon">X</font> <a href="henmodel.html#R1" target="_self">|-</a>  <a href="cqc_lang.html#K9">VERUM</a>  &amp; <font color="Maroon">X</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>  )
 
;<br/>
<b>hence </b><a NAME="E4:12"/>
not <font color="Maroon">X</font> is <a href="henmodel.html#V1" target="_self">Consistent</a>
 <b>by </b><i/>;<br/>


</div>
