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

<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> ;<br/>

<b>thus </b><a NAME="E1:11"/>
( 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">p</font> being   <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>  holds  <font color="Maroon" title="c1">X</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1"/>
not <font color="Maroon" title="c1">X</font> is <a href="henmodel.html#V1" title="HENMODEL:attr.1">Consistent</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">q</font> being    <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> <b> such that </b><br/><a NAME="E3:11_1"/><i><font color="Green" title="E10">A1</font></i>: 
( <font color="Maroon" title="c1">X</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Maroon" title="c2">q</font> &amp; <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> <font color="Maroon" title="c2">q</font> )
 <b>by </b><i><a class="ref" href="henmodel.html#D3" target="_self" title="HENMODEL:def.3">Def3</a></i>;<br/>
<b>let </b><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#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/>

<b>consider </b><font color="Maroon" title="c4">f1</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="E5:11_1"/><i><font color="Green" title="E11">A2</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f1</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="c4">f1</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"><font color="Maroon" title="c2">q</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="henmodel.html#E3:11_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="henmodel.html#D2" target="_self" title="HENMODEL:def.2">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">f2</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="E7:11_1"/><i><font color="Green" title="E12">A3</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f2</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="c5">f2</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> <font color="Maroon" title="c2">q</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="henmodel.html#E3:11_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="henmodel.html#D2" target="_self" title="HENMODEL:def.2">Def2</a></i>;<br/>
<a NAME="E8:11_1"/><i><font color="Green" title="E13">A4</font></i>: 
(  <a href="calcul_1.html#R4" title="CALCUL_1:pred.4">|-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">f2</font></span>)</span> <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"><font color="Maroon" title="c2">q</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp;  <a href="calcul_1.html#R4" title="CALCUL_1:pred.4">|-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">f2</font></span>)</span> <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> <font color="Maroon" title="c2">q</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="henmodel.html#E5:11_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="henmodel.html#E7:11_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="henmodel.html#T5" target="_self" title="HENMODEL:th.5">Th5</a>, <a class="ref" href="calcul_2.html#T20" title="CALCUL_2:th.20">CALCUL_2:20</a></i>;<br/>
<b>take </b><font color="Maroon" title="c6">f3</font> = <font color="Maroon" title="c4">f1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">f2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="henmodel.html#D2" target="_self" title="HENMODEL:def.2">HENMODEL:def 2</a><br/>

<a NAME="E9:11_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/>
<b>hence </b><a NAME="E10:11_1"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f3</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="c6">f3</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"><font color="Maroon" title="c3">p</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="henmodel.html#E5:11_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="henmodel.html#E7:11_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="henmodel.html#E8:11_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="calcul_2.html#T25" title="CALCUL_2:th.25">CALCUL_2:25</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>


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

<b>assume </b><a NAME="E2:11"/>
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#K4" title="CQC_LANG:func.4">CQC-WFF</a>  holds  <font color="Maroon" title="c1">X</font> <a href="henmodel.html#R1" title="HENMODEL:pred.1">|-</a> <font color="Olive" title="b1">p</font>
 ;<br/>

<a NAME="E3:11"/><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#K6" title="CQC_LANG:func.6">VERUM</a>  &amp; <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>  )
 
;<br/>
<b>hence </b><a NAME="E4:11"/>
not <font color="Maroon" title="c1">X</font> is <a href="henmodel.html#V1" title="HENMODEL:attr.1">Consistent</a>
 <b>by </b><i><a class="ref" href="henmodel.html#D3" target="_self" title="HENMODEL:def.3">Def3</a></i>;<br/>


</div>
