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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="funct_2.html#M2" title="FUNCT_2: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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="funct_2.html#M2" title="FUNCT_2: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/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">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/>
<span class="kw">assume </span><a NAME="E2:18"/>
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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="funct_2.html#M2" title="FUNCT_2: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="E3:18"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="henmodel.html#T6" target="_self" title="HENMODEL:th.6">Th6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">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> <span class="kw"> such that </span><br/><a NAME="E5:18"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E6:18"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
 <a href="calcul_1.html#R4" title="CALCUL_1:pred.4">|-</a> <font color="Maroon" title="c4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="henmodel.html#D2" target="_self" title="HENMODEL:def.2">Def2</a></span>;<br/>
<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font> being   <a href="funct_2.html#M2" title="FUNCT_2: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/><span class="kw">let </span><font color="Maroon" title="c6">v</font> be    <a href="funct_2.html#M2" title="FUNCT_2: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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>



<a NAME="E7:18"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
 <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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="c3">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="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="calcul_1.html#T5" title="CALCUL_1:th.5">CALCUL_1:5</a></span>;<br/>
<a NAME="E8:18"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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="c4">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="c3">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="calcul_1.html#T5" title="CALCUL_1:th.5">CALCUL_1:5</a></span>;<br/>
<a NAME="E9:18"/><span class="kw">then </span>
<font color="Maroon" title="c3">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E7:18"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="calcul_1.html#D10" title="CALCUL_1:def.10">CALCUL_1:def 10</a></span>;<br/>
<a NAME="E10:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<font color="Maroon" title="c1">X</font> <a href="calcul_1.html#R7" title="CALCUL_1:pred.7">|=</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="calcul_1.html#T32" title="CALCUL_1:th.32">CALCUL_1:32</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:18_1"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:18"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="calcul_1.html#D12" title="CALCUL_1:def.12">CALCUL_1:def 12</a></span>;<br/><a NAME="E3:18_1"/><span class="kw">then </span>
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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valuat_1.html#T28" title="VALUAT_1:th.28">VALUAT_1:28</a></span>;<br/><span class="kw">hence </span><a NAME="E4:18_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valuat_1.html#T44" title="VALUAT_1:th.44">VALUAT_1:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:18"/>
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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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