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<span class="kw">let </span><font color="Maroon" title="c1">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">psi</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="c1">S</font><br/>  for <font color="Olive" title="b2">D1</font> being   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="fomodel4.html#V12" title="FOMODEL4:attr.12">-ranked</a>  1 <a href="fomodel4.html#V12" title="FOMODEL4:attr.12">-ranked</a>  <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b3">X</font> being  <font color="Olive">b<sub>2</sub></font> <a href="fomodel4.html#V10" title="FOMODEL4:attr.10">-expanded</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">D1</font> &amp;  <a href="fomodel4.html#K29" title="FOMODEL4:func.29">R4</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">D1</font> &amp;  <a href="fomodel4.html#K31" title="FOMODEL4:func.31">R6</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">D1</font> &amp;  <a href="fomodel4.html#K32" title="FOMODEL4:func.32">R7</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">D1</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">D1</font> &amp; <font color="Olive" title="b3">X</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel2.html#V6" title="FOMODEL2:attr.6">-mincover</a>  &amp; <font color="Olive" title="b3">X</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#V15" title="FOMODEL4:attr.15">-witnessed</a>  holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b2">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Olive" title="b3">X</font></span>)</span> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Olive" title="b1">psi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Olive" title="b1">psi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">psi</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="c1">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">D1</font> being   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="fomodel4.html#V12" title="FOMODEL4:attr.12">-ranked</a>  1 <a href="fomodel4.html#V12" title="FOMODEL4:attr.12">-ranked</a>  <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b2">X</font> being  <font color="Olive">b<sub>1</sub></font> <a href="fomodel4.html#V10" title="FOMODEL4:attr.10">-expanded</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D1</font> &amp;  <a href="fomodel4.html#K29" title="FOMODEL4:func.29">R4</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D1</font> &amp;  <a href="fomodel4.html#K31" title="FOMODEL4:func.31">R6</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D1</font> &amp;  <a href="fomodel4.html#K32" title="FOMODEL4:func.32">R7</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D1</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D1</font> &amp; <font color="Olive" title="b2">X</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel2.html#V6" title="FOMODEL2:attr.6">-mincover</a>  &amp; <font color="Olive" title="b2">X</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#V15" title="FOMODEL4:attr.15">-witnessed</a>  holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Olive" title="b2">X</font></span>)</span> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c2">psi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c2">psi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">D1</font> be   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="fomodel4.html#V12" title="FOMODEL4:attr.12">-ranked</a>  1 <a href="fomodel4.html#V12" title="FOMODEL4:attr.12">-ranked</a>  <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Maroon" title="c1">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">X</font> being  <font color="Maroon" title="c3">D1</font> <a href="fomodel4.html#V10" title="FOMODEL4:attr.10">-expanded</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K29" title="FOMODEL4:func.29">R4</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K31" title="FOMODEL4:func.31">R6</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K32" title="FOMODEL4:func.32">R7</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Olive" title="b1">X</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel2.html#V6" title="FOMODEL2:attr.6">-mincover</a>  &amp; <font color="Olive" title="b1">X</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#V15" title="FOMODEL4:attr.15">-witnessed</a>  holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c3">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c2">psi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c2">psi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">X</font> be  <font color="Maroon" title="c3">D1</font> <a href="fomodel4.html#V10" title="FOMODEL4:attr.10">-expanded</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"> (  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K29" title="FOMODEL4:func.29">R4</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K31" title="FOMODEL4:func.31">R6</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K32" title="FOMODEL4:func.32">R7</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Maroon" title="c4">X</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel2.html#V6" title="FOMODEL2:attr.6">-mincover</a>  &amp; <font color="Maroon" title="c4">X</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#V15" title="FOMODEL4:attr.15">-witnessed</a>  implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c2">psi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c2">psi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> ) )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">TT</font> =  <a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">E</font> =  <a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c1">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">N</font> =  <a href="fomodel2.html#K1" title="FOMODEL2:func.1">TheNorSymbOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">R</font> = (<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">U</font> =  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">L</font> =  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">AF</font> =  <a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">d</font> = <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span></span>)</span> <a href="fomodel2.html#K2" title="FOMODEL2:func.2">-deltaInterpreter</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">i</font> = (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c4">X</font>) <a href="fomodel3.html#K7" title="FOMODEL3:func.7">-freeInterpreter</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">II</font> = <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span></span>)</span> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">D</font> = <font color="Maroon" title="c3">D1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">ii</font> = <span class="p1">(<span class="default"><a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">G0</font> =  <a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">G1</font> =  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">G2</font> =  <a href="fomodel4.html#K24" title="FOMODEL4:func.24">R2</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">G4</font> =  <a href="fomodel4.html#K29" title="FOMODEL4:func.29">R4</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">G6</font> =  <a href="fomodel4.html#K31" title="FOMODEL4:func.31">R6</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">G7</font> =  <a href="fomodel4.html#K32" title="FOMODEL4:func.32">R7</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c24">G8</font> =  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">E0</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c26">E1</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c27">E2</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K24" title="FOMODEL4:func.24">R2</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c28">E4</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K29" title="FOMODEL4:func.29">R4</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c29">E6</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K31" title="FOMODEL4:func.31">R6</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c30">E7</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K32" title="FOMODEL4:func.32">R7</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c31">E8</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c32">E0</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, <font color="Maroon" title="c33">E1</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, <font color="Maroon" title="c34">E2</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K24" title="FOMODEL4:func.24">R2</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, <font color="Maroon" title="c35">E4</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K29" title="FOMODEL4:func.29">R4</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, <font color="Maroon" title="c36">E6</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K31" title="FOMODEL4:func.31">R6</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, <font color="Maroon" title="c37">E7</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K32" title="FOMODEL4:func.32">R7</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, <font color="Maroon" title="c38">E8</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<span class="kw">assume </span><a NAME="E2:183"/>
(  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K29" title="FOMODEL4:func.29">R4</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K31" title="FOMODEL4:func.31">R6</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K32" title="FOMODEL4:func.32">R7</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> )
 ; <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="c4">X</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel2.html#V6" title="FOMODEL2:attr.6">-mincover</a>  or  not <font color="Maroon" title="c4">X</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#V15" title="FOMODEL4:attr.15">-witnessed</a>  or ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c2">psi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c2">psi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> ) )</span><br/>

<a NAME="E3:183"/><span class="kw">then </span>
(  <a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K23" title="FOMODEL4:func.23">R1</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K24" title="FOMODEL4:func.24">R2</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K29" title="FOMODEL4:func.29">R4</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K31" title="FOMODEL4:func.31">R6</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K32" title="FOMODEL4:func.32">R7</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp;  <a href="fomodel4.html#K33" title="FOMODEL4:func.33">R8</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> )
 
<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">reconsider </span><font color="Maroon" title="c39">F0</font> = <font color="Maroon" title="c32">E0</font>, <font color="Maroon" title="c40">F1</font> = <font color="Maroon" title="c33">E1</font>, <font color="Maroon" title="c41">F2</font> = <font color="Maroon" title="c34">E2</font>, <font color="Maroon" title="c42">F4</font> = <font color="Maroon" title="c35">E4</font>, <font color="Maroon" title="c43">F6</font> = <font color="Maroon" title="c36">E6</font>, <font color="Maroon" title="c44">F7</font> = <font color="Maroon" title="c37">E7</font>, <font color="Maroon" title="c45">F8</font> = <font color="Maroon" title="c38">E8</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">D1</font> <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/>
<a NAME="E5:183"/><span class="lab"><font color="Green" title="E102">B2</font></span>: 
( <font color="Maroon" title="c39">F0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c39">F0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c40">F1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c45">F8</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Maroon" title="c39">F0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c43">F6</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Maroon" title="c39">F0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Maroon" title="c39">F0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c39">F0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c40">F1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c45">F8</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c44">F7</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D1</font> )
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c46">I</font> = <font color="Maroon" title="c3">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Maroon" title="c4">X</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span></span>)</span> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c1">S</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c47">UV</font> = <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c48">uv</font> = <span class="p1">(<span class="default">(<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c4">X</font>) <a href="fomodel3.html#K7" title="FOMODEL3:func.7">-freeInterpreter</a></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c49">O</font> =  <a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c50">FF</font> =  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c51">C</font> = <font color="Maroon" title="c1">S</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c52">SS</font> =  <a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">assume </span><a NAME="E7:183"/><span class="lab"><font color="Green" title="E103">B0</font></span>: 
( <font color="Maroon" title="c4">X</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel2.html#V6" title="FOMODEL2:attr.6">-mincover</a>  &amp; <font color="Maroon" title="c4">X</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#V15" title="FOMODEL4:attr.15">-witnessed</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c2">psi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c2">psi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> 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="c1">S</font>  st <font color="Olive" title="b1">phi</font> is $1 <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  holds <br/>( <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Olive" title="b1">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Olive" title="b1">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> );<br/>
<a NAME="E8:183"/><span class="lab"><font color="Green" title="E104">B10</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="183_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c53">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="c1">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="c53">phi</font> is  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  implies ( <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c53">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c53">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:183_1"/>
<font color="Maroon" title="c53">phi</font> is  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a> 
 ; <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="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c53">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c53">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c54">phi0</font> = <font color="Maroon" title="c53">phi</font> as   <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E3:183_1"/>
( <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c54">phi0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c54">phi0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E129"><span class="lab"><font color="Green" title="E99">Lm12</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:183_1"/>
( <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c53">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c53">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )
 ; <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>
<a NAME="E9:183"/><span class="lab"><font color="Green" title="E105">B11</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="183_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c53">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c53">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c53">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c54">Vn</font> = (<font color="Maroon" title="c46">I</font>,<font color="Maroon" title="c53">n</font>) <a href="fomodel2.html#K24" title="FOMODEL2:func.24">-TruthEval</a> ;<br/>
<span class="kw">assume </span><a NAME="E1:183_2"/><span class="lab"><font color="Green" title="E106">C0</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c53">n</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">S<sub>1</sub></font>[<font color="Maroon" title="c53">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c55">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="c1">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="c55">phi</font> is <font color="Maroon" title="c53">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  implies ( <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c56">s</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c55">phi</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c57">V</font> = <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c55">phi</font>;<br/>
<span class="kw">assume </span><a NAME="E2:183_2"/><span class="lab"><font color="Green" title="E107">C1</font></span>: 
<font color="Maroon" title="c55">phi</font> is <font color="Maroon" title="c53">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a> 
 ; <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="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="183_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:183_2_1"/>
( (  not <font color="Maroon" title="c55">phi</font> is  <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a>  &amp; <font color="Maroon" title="c55">phi</font> is  <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</a>  ) or (  not <font color="Maroon" title="c55">phi</font> is  <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a>  &amp;  not <font color="Maroon" title="c55">phi</font> is  <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</a>  ) or <font color="Maroon" title="c55">phi</font> is  <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="183_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:183_2_1_1"/>
(  not <font color="Maroon" title="c55">phi</font> is  <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a>  &amp; <font color="Maroon" title="c55">phi</font> is  <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</a>  )
 ; <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="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c58">phii</font> = <font color="Maroon" title="c55">phi</font> as   non  <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a>  <font color="Maroon" title="c53">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>   <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:183_2"><span class="lab"><font color="Green" title="E107">C1</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c59">phi1</font> =  <a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font> as  <font color="Maroon" title="c53">n</font> <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c1">S</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c60">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c58">phii</font> as   <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ;<br/><span class="lab"><font color="Green" title="E108">D1</font></span>: <font color="Maroon" title="c58">phii</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c60">l</font></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="c59">phi1</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K41" title="FOMODEL2:func.41">tail</a> <font color="Maroon" title="c58">phii</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#T23" title="FOMODEL2:th.23">FOMODEL2:23</a></span>
<br/>.= 
<span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c60">l</font></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="c59">phi1</font>

;<br/><div><span class="kw">hereby </span> <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="c55">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:183_2_1_1_2"/>
<font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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="c55">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c61">u</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:183_2_1_1_2"/><span class="lab"><font color="Green" title="E109">E0</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c60">l</font>,<font color="Maroon" title="c61">u</font>) <a href="fomodel2.html#K37" title="FOMODEL2:func.37">ReassignIn</a> <font color="Maroon" title="c46">I</font></span>)</span> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c59">phi1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:183_2_1_1"><span class="lab"><font color="Green" title="E108">D1</font></span></a>, <a class="ref" href="fomodel2.html#T19" title="FOMODEL2:th.19">FOMODEL2:19</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c62">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:183_2_1_1_2"/><span class="lab"><font color="Green" title="E110">E2</font></span>: 
( <font color="Maroon" title="c62">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c61">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> (<span class="p1">(<span class="default">(<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span>,<font color="Maroon" title="c62">x</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D3" title="EQREL_1:def.3">EQREL_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c63">tt</font> = <font color="Maroon" title="c62">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:183_2_1_1_2"><span class="lab"><font color="Green" title="E110">E2</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c64">psi1</font> = (<font color="Maroon" title="c60">l</font>,<font color="Maroon" title="c63">tt</font>) <a href="fomodel3.html#K37" title="FOMODEL3:func.37">SubstIn</a> <font color="Maroon" title="c59">phi1</font> as  <font color="Maroon" title="c53">n</font> <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c1">S</font> ;<br/><a NAME="E8:183_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel2.html#K3" title="FOMODEL2:func.3">id</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c1">S</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c63">tt</font></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <font color="Maroon" title="c63">tt</font> <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="E9:183_2_1_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="fomodel2.html#K3" title="FOMODEL2:func.3">id</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c1">S</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c63">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c63">tt</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">(<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span> <a href="fomodel3.html#K17" title="FOMODEL3:func.17">-class</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">(<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c4">X</font>) <a href="fomodel3.html#K7" title="FOMODEL3:func.7">-freeInterpreter</a></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c63">tt</font></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span> <a href="fomodel3.html#K17" title="FOMODEL3:func.17">-class</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">(<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c4">X</font>) <a href="fomodel3.html#K7" title="FOMODEL3:func.7">-freeInterpreter</a></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c63">tt</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>  )
 <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="E10:183_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">E1</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c4">X</font>) <a href="fomodel3.html#K7" title="FOMODEL3:func.7">-freeInterpreter</a></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c63">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c63">tt</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span> <a href="fomodel3.html#K17" title="FOMODEL3:func.17">-class</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c4">X</font>) <a href="fomodel3.html#K7" title="FOMODEL3:func.7">-freeInterpreter</a></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c63">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span> <a href="fomodel3.html#K17" title="FOMODEL3:func.17">-class</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c4">X</font>) <a href="fomodel3.html#K7" title="FOMODEL3:func.7">-freeInterpreter</a></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c63">tt</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>, <a class="ref" href="fomodel3.html#T4" title="FOMODEL3:th.4">FOMODEL3:4</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c63">tt</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span> <a href="fomodel3.html#K17" title="FOMODEL3:func.17">-class</a></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c4">X</font>) <a href="fomodel3.html#K7" title="FOMODEL3:func.7">-freeInterpreter</a></span>)</span> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c63">tt</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel3.html#T3" title="FOMODEL3:th.3">FOMODEL3:3</a></span>
<br/>.= 
<font color="Maroon" title="c61">u</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:183_2_1_1_2"><span class="lab"><font color="Green" title="E111">E1</font></span></a>, <a class="txt" href="#E5:183_2_1_1_2"><span class="lab"><font color="Green" title="E110">E2</font></span></a>, <a class="ref" href="fomodel3.html#D13" title="FOMODEL3:def.13">FOMODEL3:def 13</a></span>
;<br/><a NAME="E12:183_2_1_1_2"/><span class="kw">then </span>
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c64">psi1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:183_2_1_1_2"><span class="lab"><font color="Green" title="E109">E0</font></span></a>, <a class="ref" href="fomodel3.html#T10" title="FOMODEL3:th.10">FOMODEL3:10</a></span>;<br/><a NAME="E13:183_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c64">psi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:183_2"><span class="lab"><font color="Green" title="E106">C0</font></span></a></span>;<br/><a NAME="E14:183_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">E3</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c64">psi1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 <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/><a NAME="E15:183_2_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c60">l</font>,<font color="Maroon" title="c63">tt</font>) <a href="fomodel3.html#K37" title="FOMODEL3:func.37">SubstIn</a> <font color="Maroon" title="c59">phi1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2">(<span class="default"><span class="p3"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c60">l</font></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="c59">phi1</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is 1, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K29" title="FOMODEL4:func.29">R4</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V4" title="FOMODEL4:attr.4">-derivable</a> 
 ;<br/><a NAME="E16:183_2_1_1_2"/><span class="kw">then </span>
( <span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><font color="Maroon" title="c60">l</font></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="c59">phi1</font> is <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c35">E4</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a>  &amp; <font color="Maroon" title="c42">F4</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Maroon" title="c35">E4</font> is  <a href="fomodel4.html#V6" title="FOMODEL4:attr.6">isotone</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:183_2_1_1_2"><span class="lab"><font color="Green" title="E112">E3</font></span></a>, <a class="ref" href="fomodel4.html#D62" target="_self" title="FOMODEL4:def.62">DefProvable3</a></span>;<br/><a NAME="E17:183_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c58">phii</font> is <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:183_2_1_1"><span class="lab"><font color="Green" title="E108">D1</font></span></a>, <a class="txt" href="fomodel4.html#E110"><span class="lab"><font color="Green" title="E89">Lm35</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E18:183_2_1_1_2"/>
<font color="Maroon" title="c55">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D61" target="_self" title="FOMODEL4:def.61">DefExpanded2</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">assume </span><a NAME="E7:183_2_1_1"/>
<font color="Maroon" title="c55">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c61">l2</font> being   <a href="fomodel1.html#V4" title="FOMODEL1:attr.4">literal</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E9:183_2_1_1"/><span class="lab"><font color="Green" title="E109">E0</font></span>: 
( (<font color="Maroon" title="c60">l</font>,<font color="Maroon" title="c61">l2</font>) <a href="fomodel3.html#K24" title="FOMODEL3:func.24">-SymbolSubstIn</a> <font color="Maroon" title="c59">phi1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp;  not <font color="Maroon" title="c61">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c59">phi1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:183"><span class="lab"><font color="Green" title="E103">B0</font></span></a>, <a class="ref" href="fomodel4.html#D66" target="_self" title="FOMODEL4:def.66">DefWit1</a>, <a class="txt" href="#E5:183_2_1_1"><span class="lab"><font color="Green" title="E108">D1</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c62">psi1</font> = (<font color="Maroon" title="c60">l</font>,<font color="Maroon" title="c61">l2</font>) <a href="fomodel3.html#K24" title="FOMODEL3:func.24">-SymbolSubstIn</a> <font color="Maroon" title="c59">phi1</font> as  <font color="Maroon" title="c53">n</font> <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c1">S</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c63">u</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E12:183_2_1_1"/><span class="lab"><font color="Green" title="E110">E1</font></span>: 
( <font color="Maroon" title="c63">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c61">l2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> &amp; (<font color="Maroon" title="c61">l2</font>,<font color="Maroon" title="c63">u</font>) <a href="fomodel2.html#K37" title="FOMODEL2:func.37">ReassignIn</a> <font color="Maroon" title="c46">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c46">I</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#T26" title="FOMODEL2:th.26">FOMODEL2:26</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c64">I2</font> = (<font color="Maroon" title="c61">l2</font>,<font color="Maroon" title="c63">u</font>) <a href="fomodel2.html#K37" title="FOMODEL2:func.37">ReassignIn</a> <font color="Maroon" title="c46">I</font>, <font color="Maroon" title="c65">I1</font> = (<font color="Maroon" title="c60">l</font>,<font color="Maroon" title="c63">u</font>) <a href="fomodel2.html#K37" title="FOMODEL2:func.37">ReassignIn</a> <font color="Maroon" title="c46">I</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p2">(<span class="default">(<font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span></span>)</span> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c1">S</font> ;<br/><a NAME="E14:183_2_1_1"/>
<font color="Maroon" title="c64">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c62">psi1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:183_2_1_1"><span class="lab"><font color="Green" title="E109">E0</font></span></a>, <a class="txt" href="#E1:183_2"><span class="lab"><font color="Green" title="E106">C0</font></span></a>, <a class="txt" href="#E12:183_2_1_1"><span class="lab"><font color="Green" title="E110">E1</font></span></a></span>;<br/><a NAME="E15:183_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c65">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c59">phi1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:183_2_1_1"><span class="lab"><font color="Green" title="E109">E0</font></span></a>, <a class="ref" href="fomodel3.html#T9" title="FOMODEL3:th.9">FOMODEL3:9</a></span>;<br/><span class="kw">hence </span><a NAME="E16:183_2_1_1"/>
<font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:183_2_1_1"><span class="lab"><font color="Green" title="E108">D1</font></span></a>, <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="183_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:183_2_1_2"/>
(  not <font color="Maroon" title="c55">phi</font> is  <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a>  &amp;  not <font color="Maroon" title="c55">phi</font> is  <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</a>  )
 ; <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="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c58">phii</font> = <font color="Maroon" title="c55">phi</font> as   non  <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a>  <font color="Maroon" title="c53">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>   non  <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:183_2"><span class="lab"><font color="Green" title="E107">C1</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c59">phi1</font> =  <a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font>;<br/><span class="kw">set </span><font color="Maroon" title="c60">phi2</font> =  <a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font>;<br/><a NAME="E3:183_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c58">phii</font></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K1" title="FOMODEL2:func.1">TheNorSymbOf</a> <font color="Maroon" title="c1">S</font></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="E4:183_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel2.html#K1" title="FOMODEL2:func.1">TheNorSymbOf</a> <font color="Maroon" title="c1">S</font>
 <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="E5:183_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">D0</font></span>: 
<font color="Maroon" title="c55">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c1">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> <span class="p2">(<span class="default"><a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font></span>)</span></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#T23" title="FOMODEL2:th.23">FOMODEL2:23</a></span>;<br/><a NAME="E6:183_2_1_2"/>
( <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff ( <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:183_2_1_2"><span class="lab"><font color="Green" title="E108">D0</font></span></a>, <a class="ref" href="fomodel2.html#T19" title="FOMODEL2:th.19">FOMODEL2:19</a></span>;<br/><a NAME="E7:183_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff (  not <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  not <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T39" title="FOMODEL0:th.39">FOMODEL0:39</a></span>;<br/><a NAME="E8:183_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E109">DD2</font></span>: 
( <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff (  not  <a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp;  not  <a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:183_2"><span class="lab"><font color="Green" title="E106">C0</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E110">DD3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="183_2_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:183_2_1_2_1"/>
(  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="c55">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font></span><br/><a NAME="E2:183_2_1_2_1"/><span class="kw">then </span>
(  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font></span>)</span> is <font color="Maroon" title="c4">X</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a>  &amp;  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font></span>)</span> is <font color="Maroon" title="c4">X</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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#T4" target="_self" title="FOMODEL4:th.4">Lm2</a></span>;<br/><a NAME="E3:183_2_1_2_1"/><span class="kw">then </span>
(  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font></span>)</span> is <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a>  &amp;  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font></span>)</span> is <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:183"><span class="lab"><font color="Green" title="E102">B2</font></span></a>, <a class="txt" href="fomodel4.html#E110"><span class="lab"><font color="Green" title="E89">Lm35</font></span></a></span>;<br/><a NAME="E4:183_2_1_2_1"/><span class="kw">then </span>
(  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D61" target="_self" title="FOMODEL4:def.61">DefExpanded2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c61">Y</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p3">(<span class="default"><a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p3">(<span class="default"><a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font></span>)</span></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/><a NAME="E6:183_2_1_2_1"/>
<font color="Maroon" title="c55">phi</font> is <font color="Maroon" title="c4">X</font> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> <font color="Maroon" title="c61">Y</font>,<font color="Maroon" title="c3">D1</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E110"><span class="lab"><font color="Green" title="E89">Lm35</font></span></a>, <a class="txt" href="#E5:183"><span class="lab"><font color="Green" title="E102">B2</font></span></a>, <a class="txt" href="#E5:183_2_1_2"><span class="lab"><font color="Green" title="E108">D0</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:183_2_1_2_1"/>
<font color="Maroon" title="c55">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D61" target="_self" title="FOMODEL4:def.61">DefExpanded2</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="183_2_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c61">H</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c55">phi</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as  <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#V13" title="FOMODEL4:attr.13">-premises-like</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">assume </span><a NAME="E2:183_2_1_2_2"/>
<font color="Maroon" title="c55">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <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> <span class="p1">(<span class="default"><a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp;  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )</span><br/><a NAME="E3:183_2_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">E7</font></span>: 
<font color="Maroon" title="c61">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 <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/><a NAME="E4:183_2_1_2_2"/><span class="lab"><font color="Green" title="E112">E3</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c55">phi</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c55">phi</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is 1, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c32">E0</font> <a href="fomodel4.html#V4" title="FOMODEL4:attr.4">-derivable</a> 
 ;<br/><a NAME="E5:183_2_1_2_2"/><span class="lab"><font color="Green" title="E113">EE0</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c61">H</font> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> <span class="p3">(<span class="default"><a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p3">(<span class="default"><a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is 2,<span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c55">phi</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c55">phi</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="lang1.html#K4" title="LANG1:func.4">}</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c32">E0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c33">E1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c38">E8</font> <a href="fomodel4.html#V4" title="FOMODEL4:attr.4">-derivable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:183_2_1_2"><span class="lab"><font color="Green" title="E108">D0</font></span></a></span>;<br/><a NAME="E6:183_2_1_2_2"/><span class="lab"><font color="Green" title="E114">EE1</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c61">H</font> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> <span class="p3">(<span class="default"><a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p3">(<span class="default"><a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is 3,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c61">H</font>,<font color="Maroon" title="c55">phi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">E0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c33">E1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c38">E8</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c37">E7</font> <a href="fomodel4.html#V4" title="FOMODEL4:attr.4">-derivable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:183_2_1_2"><span class="lab"><font color="Green" title="E108">D0</font></span></a></span>;<br/><a NAME="E7:183_2_1_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c61">H</font>,<span class="p2">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p3">(<span class="default"><a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 2, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c32">E0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">E0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c33">E1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c38">E8</font></span>)</span> <a href="fomodel4.html#V4" title="FOMODEL4:attr.4">-derivable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:183_2_1_2_2"><span class="lab"><font color="Green" title="E113">EE0</font></span></a>, <a class="txt" href="fomodel4.html#E38"><span class="lab"><font color="Green" title="E33">Lm28</font></span></a></span>;<br/><a NAME="E8:183_2_1_2_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c61">H</font>,<span class="p2">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p3">(<span class="default"><a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is 3, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c3">D1</font> <a href="fomodel4.html#V4" title="FOMODEL4:attr.4">-derivable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:183"><span class="lab"><font color="Green" title="E102">B2</font></span></a>, <a class="ref" href="fomodel4.html#T11" target="_self" title="FOMODEL4:th.11">Th12</a></span>;<br/><a NAME="E9:183_2_1_2_2"/><span class="kw">then </span>
 <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font></span>)</span> is <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font> <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#D62" target="_self" title="FOMODEL4:def.62">DefProvable3</a>, <a class="txt" href="#E3:183_2_1_2_2"><span class="lab"><font color="Green" title="E111">E7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:183_2_1_2_2"/>
 <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c58">phii</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D61" target="_self" title="FOMODEL4:def.61">DefExpanded2</a></span>; <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> <span class="p1">(<span class="default"><a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font></span><br/><a NAME="E11:183_2_1_2_2"/><span class="lab"><font color="Green" title="E115">EE8</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c61">H</font>,<span class="p2">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p3">(<span class="default"><a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 3, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c32">E0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c32">E0</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c33">E1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c38">E8</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c37">E7</font></span>)</span> <a href="fomodel4.html#V4" title="FOMODEL4:attr.4">-derivable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:183_2_1_2_2"><span class="lab"><font color="Green" title="E112">E3</font></span></a>, <a class="txt" href="#E6:183_2_1_2_2"><span class="lab"><font color="Green" title="E114">EE1</font></span></a>, <a class="txt" href="fomodel4.html#E38"><span class="lab"><font color="Green" title="E33">Lm28</font></span></a></span>;<br/><a NAME="E12:183_2_1_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c61">H</font>,<span class="p2">(<span class="default"><a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p3">(<span class="default"><a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is 4, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c3">D1</font> <a href="fomodel4.html#V4" title="FOMODEL4:attr.4">-derivable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:183"><span class="lab"><font color="Green" title="E102">B2</font></span></a>, <a class="txt" href="#E11:183_2_1_2_2"><span class="lab"><font color="Green" title="E115">EE8</font></span></a>, <a class="ref" href="fomodel4.html#T11" target="_self" title="FOMODEL4:th.11">Th12</a></span>;<br/><a NAME="E13:183_2_1_2_2"/><span class="kw">then </span>
 <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font></span>)</span> is <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c3">D1</font> <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#D62" target="_self" title="FOMODEL4:def.62">DefProvable3</a>, <a class="txt" href="#E3:183_2_1_2_2"><span class="lab"><font color="Green" title="E111">E7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E14:183_2_1_2_2"/>
 <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c58">phii</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D61" target="_self" title="FOMODEL4:def.61">DefExpanded2</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="E11:183_2_1_2"/>
( <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:183_2_1_2"><span class="lab"><font color="Green" title="E109">DD2</font></span></a>, <a class="txt" href="#E7:183"><span class="lab"><font color="Green" title="E103">B0</font></span></a>, <a class="txt" href="#E9:183_2_1_2"><span class="lab"><font color="Green" title="E110">DD3</font></span></a>, <a class="ref" href="fomodel2.html#D34" title="FOMODEL2:def.34">FOMODEL2:def 34</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="183_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:183_2_1_3"/>
<font color="Maroon" title="c55">phi</font> is  <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a> 
 ; <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="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:183_2_1_3"/>
( <font color="Maroon" title="c46">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c55">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:183"><span class="lab"><font color="Green" title="E104">B10</font></span></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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E10:183"/><span class="lab"><font color="Green" title="E106">B12</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E8:183"><span class="lab"><font color="Green" title="E104">B10</font></span></a>, <a class="txt" href="#E9:183"><span class="lab"><font color="Green" title="E105">B11</font></span></a>);<br/></span>
<a NAME="E11:183"/>
<font color="Maroon" title="c2">psi</font> is  <a href="fomodel2.html#K32" title="FOMODEL2:func.32">Depth</a> <font color="Maroon" title="c2">psi</font> <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#D31" title="FOMODEL2:def.31">FOMODEL2:def 31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:183"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c2">psi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c2">psi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:183"><span class="lab"><font color="Green" title="E106">B12</font></span></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>
