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<span class="kw">let </span><font color="Maroon" title="c1">Z</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="orders_4.html#V1" title="ORDERS_4:attr.1">countable</a>  <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a><br/>  for <font color="Olive" title="b2">phi</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="Olive" title="b1">S</font>  st <font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Olive" title="b1">S</font> &amp;  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Olive" title="b2">phi</font> is <font color="Maroon" title="c1">Z</font> <a href="fomodel2.html#V13" title="FOMODEL2:attr.13">-implied</a>  holds <br/> <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Olive" title="b2">phi</font> is <font color="Maroon" title="c1">Z</font>,<font color="Olive" title="b1">S</font> <a href="fomodel4.html#K37" title="FOMODEL4:func.37">-rules</a>  <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="orders_4.html#V1" title="ORDERS_4:attr.1">countable</a>  <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">phi</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="c2">S</font>  st <font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Olive" title="b1">phi</font> is <font color="Maroon" title="c1">Z</font> <a href="fomodel2.html#V13" title="FOMODEL2:attr.13">-implied</a>  holds <br/> <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Olive" title="b1">phi</font> is <font color="Maroon" title="c1">Z</font>,<font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K37" title="FOMODEL4:func.37">-rules</a>  <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a> </span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">D</font> = <font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K37" title="FOMODEL4:func.37">-rules</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">FF</font> =  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">phi</font> be   <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="c2">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="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c2">S</font> &amp;  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c5">phi</font> is <font color="Maroon" title="c1">Z</font> <a href="fomodel2.html#V13" title="FOMODEL2:attr.13">-implied</a>  implies  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c5">phi</font> is <font color="Maroon" title="c1">Z</font>,<font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K37" title="FOMODEL4:func.37">-rules</a>  <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:262"/>
<font color="Maroon" title="c1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c5">phi</font> is <font color="Maroon" title="c1">Z</font> <a href="fomodel2.html#V13" title="FOMODEL2:attr.13">-implied</a>  or  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c5">phi</font> is <font color="Maroon" title="c1">Z</font>,<font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K37" title="FOMODEL4:func.37">-rules</a>  <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a>  )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">X</font> = <font color="Maroon" title="c1">Z</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c2">S</font></span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">psi</font> =  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c5">phi</font>;<br/>
<a NAME="E3:262"/>
<font color="Maroon" title="c5">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#T16" title="FOMODEL2:th.16">FOMODEL2:16</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">Phi</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">phi</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">Y</font> = <font color="Maroon" title="c6">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c8">Phi</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c2">S</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">XX</font> = <font color="Maroon" title="c6">X</font> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> <font color="Maroon" title="c8">Phi</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c9">Y</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">Phii</font> = <font color="Maroon" title="c8">Phi</font> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> <font color="Maroon" title="c6">X</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c9">Y</font> ;<br/>
<span class="kw">assume </span><a NAME="E8:262"/><span class="lab"><font color="Green" title="E141">B2</font></span>: 
 <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c5">phi</font> is <font color="Maroon" title="c1">Z</font> <a href="fomodel2.html#V13" title="FOMODEL2:attr.13">-implied</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c5">phi</font> is <font color="Maroon" title="c1">Z</font>,<font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K37" title="FOMODEL4:func.37">-rules</a>  <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a> </span><br/>

<span class="kw">assume </span><a NAME="E9:262"/>
 not  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c5">phi</font> is <font color="Maroon" title="c1">Z</font>,<font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K37" title="FOMODEL4:func.37">-rules</a>  <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/>

<a NAME="E10:262"/><span class="kw">then </span>
( <font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K37" title="FOMODEL4:func.37">-rules</a>  is  <a href="fomodel4.html#V6" title="FOMODEL4:attr.6">isotone</a>  &amp;  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K37" title="FOMODEL4:func.37">-rules</a>  &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K37" title="FOMODEL4:func.37">-rules</a>  &amp;  not  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c5">phi</font> is <font color="Maroon" title="c1">Z</font>,<font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K37" title="FOMODEL4:func.37">-rules</a>  <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D59" target="_self" title="FOMODEL4:def.59">DefRank</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">U</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> , <font color="Maroon" title="c13">I</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c12">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E12:262"/><span class="lab"><font color="Green" title="E142">B0</font></span>: 
<font color="Maroon" title="c9">Y</font> is <font color="Maroon" title="c13">I</font> <a href="fomodel2.html#V10" title="FOMODEL2:attr.10">-satisfied</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#T20" target="_self" title="FOMODEL4:th.20">Th21</a>, <a class="txt" href="fomodel4.html#E135"><span class="lab"><font color="Green" title="E105">Lm2b</font></span></a></span>;<br/>
<a NAME="E13:262"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <span class="p2">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c5">phi</font></span>)</span></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <span class="p1">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c5">phi</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E14:262"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">B3</font></span>: 
<font color="Maroon" title="c13">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c5">phi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c5">phi</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T29" title="FOMODEL0:th.29">FOMODEL0:29</a></span>;<br/>
<a NAME="E15:262"/><span class="lab"><font color="Green" title="E144">BB1</font></span>: 
<font color="Maroon" title="c9">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">XX</font> is <font color="Maroon" title="c13">I</font> <a href="fomodel2.html#V10" title="FOMODEL2:attr.10">-satisfied</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:262"><span class="lab"><font color="Green" title="E142">B0</font></span></a></span>;<br/>
<a NAME="E16:262"/>
<font color="Maroon" title="c5">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Phii</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E17:262"/><span class="kw">then </span>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <span class="p2">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c5">phi</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:262"><span class="lab"><font color="Green" title="E143">B3</font></span></a>, <a class="txt" href="#E12:262"><span class="lab"><font color="Green" title="E142">B0</font></span></a>, <a class="ref" href="fomodel2.html#D42" title="FOMODEL2:def.42">FOMODEL2:def 42</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:262"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:262"><span class="lab"><font color="Green" title="E144">BB1</font></span></a>, <a class="txt" href="#E8:262"><span class="lab"><font color="Green" title="E141">B2</font></span></a>, <a class="ref" href="fomodel2.html#D45" title="FOMODEL2:def.45">FOMODEL2:def 45</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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