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<span class="kw">let </span><font color="Maroon" title="c1">U</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">S</font> being   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a><br/>  for <font color="Olive" title="b3">D</font> being   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Olive" title="b2">S</font><br/>  for <font color="Olive" title="b4">I</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Olive" title="b2">S</font>  st <font color="Olive" title="b3">D</font> is  <a href="fomodel4.html#V16" title="FOMODEL4:attr.16">Correct</a>  &amp; <font color="Olive" title="b1">X</font> is <font color="Olive" title="b4">I</font> <a href="fomodel2.html#V10" title="FOMODEL2:attr.10">-satisfied</a>  holds <br/> not <font color="Olive" title="b1">X</font> is <font color="Olive" title="b3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a><br/>  for <font color="Olive" title="b2">D</font> being   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">I</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b2">D</font> is  <a href="fomodel4.html#V16" title="FOMODEL4:attr.16">Correct</a>  &amp; <font color="Maroon" title="c2">X</font> is <font color="Olive" title="b3">I</font> <a href="fomodel2.html#V10" title="FOMODEL2:attr.10">-satisfied</a>  holds <br/> not <font color="Maroon" title="c2">X</font> is <font color="Olive" title="b2">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font> be   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">D</font> being   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Maroon" title="c3">S</font><br/>  for <font color="Olive" title="b2">I</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c3">S</font>  st <font color="Olive" title="b1">D</font> is  <a href="fomodel4.html#V16" title="FOMODEL4:attr.16">Correct</a>  &amp; <font color="Maroon" title="c2">X</font> is <font color="Olive" title="b2">I</font> <a href="fomodel2.html#V10" title="FOMODEL2:attr.10">-satisfied</a>  holds <br/> not <font color="Maroon" title="c2">X</font> is <font color="Olive" title="b1">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">D</font> be   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Maroon" title="c3">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">I</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c3">S</font>  st <font color="Maroon" title="c4">D</font> is  <a href="fomodel4.html#V16" title="FOMODEL4:attr.16">Correct</a>  &amp; <font color="Maroon" title="c2">X</font> is <font color="Olive" title="b1">I</font> <a href="fomodel2.html#V10" title="FOMODEL2:attr.10">-satisfied</a>  holds <br/> not <font color="Maroon" title="c2">X</font> is <font color="Maroon" title="c4">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> </span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">N</font> =  <a href="fomodel2.html#K1" title="FOMODEL2:func.1">TheNorSymbOf</a> <font color="Maroon" title="c3">S</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">I</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c3">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">D</font> is  <a href="fomodel4.html#V16" title="FOMODEL4:attr.16">Correct</a>  &amp; <font color="Maroon" title="c2">X</font> is <font color="Maroon" title="c6">I</font> <a href="fomodel2.html#V10" title="FOMODEL2:attr.10">-satisfied</a>  implies  not <font color="Maroon" title="c2">X</font> is <font color="Maroon" title="c4">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:195"/><span class="lab"><font color="Green" title="E110">B0</font></span>: 
( <font color="Maroon" title="c4">D</font> is  <a href="fomodel4.html#V16" title="FOMODEL4:attr.16">Correct</a>  &amp; <font color="Maroon" title="c2">X</font> is <font color="Maroon" title="c6">I</font> <a href="fomodel2.html#V10" title="FOMODEL2:attr.10">-satisfied</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c2">X</font> is <font color="Maroon" title="c4">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="195_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c7">phi1</font>, <font color="Maroon" title="c8">phi2</font> being   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c3">S</font><span class="kw"> such that </span><a NAME="E2:195_1"/><span class="lab"><font color="Green" title="E111">C0</font></span>: 
( <font color="Maroon" title="c7">phi1</font> is <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c4">D</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a>  &amp; <span class="p1">(<span class="default"><span class="p2"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="fomodel2.html#K1" title="FOMODEL2:func.1">TheNorSymbOf</a> <font color="Maroon" title="c3">S</font></span>)</span></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c7">phi1</font></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c8">phi2</font> is <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c4">D</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">set </span><font color="Maroon" title="c9">nphi1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="fomodel2.html#K1" title="FOMODEL2:func.1">TheNorSymbOf</a> <font color="Maroon" title="c3">S</font></span>)</span></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c7">phi1</font></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c8">phi2</font>;<br/><a NAME="E3:195_1"/>
( <font color="Maroon" title="c6">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c7">phi1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c6">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="fomodel2.html#K1" title="FOMODEL2:func.1">TheNorSymbOf</a> <font color="Maroon" title="c3">S</font></span>)</span></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c7">phi1</font></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c8">phi2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:195_1"><span class="lab"><font color="Green" title="E111">C0</font></span></a>, <a class="txt" href="#E1:195"><span class="lab"><font color="Green" title="E110">B0</font></span></a>, <a class="ref" href="fomodel4.html#D68" target="_self" title="FOMODEL4:def.68">DefCorrect</a></span>;<br/><span class="kw">hence </span><a NAME="E4:195_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#T19" title="FOMODEL2:th.19">FOMODEL2:19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:195"/>
 not <font color="Maroon" title="c2">X</font> is <font color="Maroon" title="c4">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D65" target="_self" title="FOMODEL4:def.65">DefInc</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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