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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">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> <br/>  for <font color="Olive" title="b3">phi</font> being   <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> 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#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Olive" title="b3">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Olive" title="b3">phi</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="c2">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="c2">D1</font> <a href="fomodel4.html#V10" title="FOMODEL4:attr.10">-expanded</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">phi</font> being   <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> holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">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#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Olive" title="b2">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Olive" title="b2">phi</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="c3">X</font> be  <font color="Maroon" title="c2">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">  for <font color="Olive" title="b1">phi</font> being   <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> holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</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="c3">X</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">phi</font> be   <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>; <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="c2">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )</span><br/>







<a NAME="E1:181"/>
 <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="c2">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/>
<a NAME="E2:181"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">B10</font></span>: 
<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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">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/>
<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">p</font> =  <a href="fomodel1.html#K34" title="FOMODEL1:func.34">SubTerms</a> <font color="Maroon" title="c4">phi</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">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="c9">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="c4">phi</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">n</font> =  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p2">(<span class="default"><span class="p3">(<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="c4">phi</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">R</font> = (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">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="c3">X</font>,<font color="Maroon" title="c2">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="c13">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="c14">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="c3">X</font>,<font color="Maroon" title="c2">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="c15">i</font> = (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">X</font>) <a href="fomodel3.html#K7" title="FOMODEL3:func.7">-freeInterpreter</a> ;<br/>
<a NAME="E3:181"/><span class="lab"><font color="Green" title="E101">BB0</font></span>: 
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p3">(<span class="default"><a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">I</font> = <font color="Maroon" title="c2">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Maroon" title="c3">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="c3">X</font>,<font color="Maroon" title="c2">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="c17">UV</font> = <font color="Maroon" title="c16">I</font> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">V</font> = <font color="Maroon" title="c16">I</font> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">uv</font> = <span class="p1">(<span class="default">(<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">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="c20">v</font> = <span class="p1">(<span class="default">(<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">X</font>) <a href="fomodel3.html#K7" title="FOMODEL3:func.7">-freeInterpreter</a></span>)</span> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c16">I</font> <a href="fomodel2.html#K11" title="FOMODEL2:func.11">===</a></span>)</span> <a href="fomodel2.html#K12" title="FOMODEL2:func.12">.</a> <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="c4">phi</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">G</font> = <font color="Maroon" title="c16">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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="c4">phi</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">g</font> = <span class="p1">(<span class="default">(<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">X</font>) <a href="fomodel3.html#K7" title="FOMODEL3:func.7">-freeInterpreter</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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="c4">phi</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c24">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="c25">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="c26">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="c27">SS</font> =  <a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c28">pp</font> =  <a href="fomodel1.html#K34" title="FOMODEL1:func.34">SubTerms</a> <font color="Maroon" title="c4">phi</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p3">(<span class="default"><span class="p4">(<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="c4">phi</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T16" title="FOMODEL0:th.16">FOMODEL0:16</a></span>;<br/>
<a NAME="E6:181"/>
<font color="Maroon" title="c28">pp</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p4">(<span class="default"><span class="p5">(<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="c4">phi</font></span>)</span></span>)</span></span>)</span></span>)</span>,<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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T11" title="FOMODEL0:th.11">FOMODEL0:11</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">fp</font> = <font color="Maroon" title="c28">pp</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p4">(<span class="default"><span class="p5">(<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="c4">phi</font></span>)</span></span>)</span></span>)</span></span>)</span>,<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> ;<br/>
<a NAME="E8:181"/><span class="lab"><font color="Green" title="E102">B4</font></span>: 
2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">tt1</font>,<font color="Olive" title="b2">tt2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> where <font color="Olive" title="b1">tt1</font>, <font color="Olive" title="b2">tt2</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> : verum </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T99" title="FINSEQ_2:th.99">FINSEQ_2:99</a></span>;<br/>
<a NAME="E9:181"/>
 <a href="fomodel1.html#K34" title="FOMODEL1:func.34">SubTerms</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="fomodel0.html#K2" title="FOMODEL0:func.2">*</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">Pp</font> =  <a href="fomodel1.html#K34" title="FOMODEL1:func.34">SubTerms</a> <font color="Maroon" title="c4">phi</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a>  ;<br/>
<a NAME="E11:181"/><span class="lab"><font color="Green" title="E103">B5</font></span>: 
<font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c4">phi</font></span>)</span></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K34" title="FOMODEL1:func.34">SubTerms</a> <font color="Maroon" title="c4">phi</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#D38" title="FOMODEL1:def.38">FOMODEL1:def 38</a></span>;<br/>
<a NAME="E12:181"/><span class="lab"><font color="Green" title="E104">B1</font></span>: 
<font color="Maroon" title="c16">I</font> <a href="fomodel2.html#K10" title="FOMODEL2:func.10">-TermEval</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">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="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">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 class="kw">by</span> <span class="lab"><a class="ref" href="fomodel3.html#T3" title="FOMODEL3:th.3">FOMODEL3:3</a></span>;<br/>
<a NAME="E13:181"/><span class="lab"><font color="Green" title="E105">B2</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">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>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="fomodel2.html#K3" title="FOMODEL2:func.3">id</a> <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel3.html#T4" title="FOMODEL3:th.4">FOMODEL3:4</a></span>;<br/>
<a NAME="E14:181"/>
( <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p3">(<span class="default"><span class="p4">(<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="c4">phi</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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="fomodel0.html#K2" title="FOMODEL0:func.2">*</a>  &amp; <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="fomodel0.html#K2" title="FOMODEL0:func.2">*</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T142" title="FINSEQ_2:th.142">FINSEQ_2:142</a></span>;<br/>
<a NAME="E15:181"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p3">(<span class="default"><span class="p4">(<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="c4">phi</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">nc</font> = <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> <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="c4">phi</font></span>)</span> <a href="fomodel3.html#K2" title="FOMODEL3:func.2">-compound</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p4">(<span class="default"><span class="p5">(<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="c4">phi</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</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> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p4">(<span class="default"><span class="p5">(<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="c4">phi</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</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>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T32" title="FUNCT_2:th.32">FUNCT_2:32</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="181_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="E1:181_1"/>
( <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="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font> or  not <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="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="181_1_1"><span class="kw">suppose </span></a><a NAME="E1:181_1_1"/><span class="lab"><font color="Green" title="E106">C6</font></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="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <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"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c32">p1</font> =  <a href="fomodel1.html#K34" title="FOMODEL1:func.34">SubTerms</a> <font color="Maroon" title="c4">phi</font> as  <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <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="fomodel0.html#K2" title="FOMODEL0:func.2">*</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:181_1_1"><span class="lab"><font color="Green" title="E106">C6</font></span></a>, <a class="txt" href="#E3:181"><span class="lab"><font color="Green" title="E101">BB0</font></span></a></span>;<br/><a NAME="E3:181_1_1"/>
<font color="Maroon" title="c30">Pp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:181"><span class="lab"><font color="Green" title="E101">BB0</font></span></a>, <a class="txt" href="#E1:181_1_1"><span class="lab"><font color="Green" title="E106">C6</font></span></a>, <a class="ref" href="fomodel0.html#T16" title="FOMODEL0:th.16">FOMODEL0:16</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c33">tt11</font>, <font color="Maroon" title="c34">tt22</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span><span class="kw"> such that </span><br/><a NAME="E5:181_1_1"/><span class="lab"><font color="Green" title="E107">C3</font></span>: 
<font color="Maroon" title="c30">Pp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c33">tt11</font>,<font color="Maroon" title="c34">tt22</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:181"><span class="lab"><font color="Green" title="E102">B4</font></span></a></span>;<br/><a NAME="E6:181_1_1"/><span class="lab"><font color="Green" title="E108">C4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c33">tt11</font>,<font color="Maroon" title="c34">tt22</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">tt11</font> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c34">tt22</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T15" title="FOMODEL0:th.15">FOMODEL0:15</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c35">p2</font> =  <a href="fomodel1.html#K34" title="FOMODEL1:func.34">SubTerms</a> <font color="Maroon" title="c4">phi</font> as  <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <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="fomodel0.html#K2" title="FOMODEL0:func.2">*</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:181_1_1"><span class="lab"><font color="Green" title="E106">C6</font></span></a>, <a class="txt" href="#E3:181"><span class="lab"><font color="Green" title="E101">BB0</font></span></a></span>;<br/><a NAME="E8:181_1_1"/>
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c35">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c36">tt1</font> = <span class="p1">(<span class="default"><a href="fomodel1.html#K34" title="FOMODEL1:func.34">SubTerms</a> <font color="Maroon" title="c4">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1, <font color="Maroon" title="c37">tt2</font> = <span class="p1">(<span class="default"><a href="fomodel1.html#K34" title="FOMODEL1:func.34">SubTerms</a> <font color="Maroon" title="c4">phi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 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="ref" href="zfmisc_1.html#T60" title="ZFMISC_1:th.60">ZFMISC_1:60</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c38">t1</font> = <font color="Maroon" title="c36">tt1</font>, <font color="Maroon" title="c39">t2</font> = <font color="Maroon" title="c37">tt2</font> as   <a href="fomodel1.html#V13" title="FOMODEL1:attr.13">termal</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c1">S</font> ;<br/><a NAME="E11:181_1_1"/><span class="lab"><font color="Green" title="E109">C8</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">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> <font color="Maroon" title="c36">tt1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> (<span class="p1">(<span class="default">(<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span>,<font color="Maroon" title="c36">tt1</font>) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">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> <font color="Maroon" title="c37">tt2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> (<span class="p1">(<span class="default">(<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span>,<font color="Maroon" title="c37">tt2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel3.html#D13" title="FOMODEL3:def.13">FOMODEL3:def 13</a></span>;<br/><a NAME="E12:181_1_1"/><span class="lab"><font color="Green" title="E110">C5</font></span>: 
( <font color="Maroon" title="c36">tt1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">tt11</font> &amp; <font color="Maroon" title="c37">tt2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">tt22</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:181_1_1"><span class="lab"><font color="Green" title="E107">C3</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><a NAME="E13:181_1_1"/>
( <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel2.html#K3" title="FOMODEL2:func.3">id</a> <span class="p4">(<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="c36">tt1</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c36">tt1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel2.html#K3" title="FOMODEL2:func.3">id</a> <span class="p4">(<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="c37">tt2</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c37">tt2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/><a NAME="E14:181_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E111">C1</font></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="c36">tt1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">tt1</font> &amp; <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="c37">tt2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">tt2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T15" title="ZFMISC_1:th.15">ZFMISC_1:15</a></span>;<br/><a NAME="E15:181_1_1"/>
( <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="c3">X</font>,<font color="Maroon" title="c2">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="c3">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="c36">tt1</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="c3">X</font>,<font color="Maroon" title="c2">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="c3">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="c36">tt1</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>  &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="c3">X</font>,<font color="Maroon" title="c2">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="c3">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="c37">tt2</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="c3">X</font>,<font color="Maroon" title="c2">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="c3">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="c37">tt2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/><a NAME="E16:181_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">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="c3">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="c36">tt1</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="c3">X</font>,<font color="Maroon" title="c2">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="c3">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="c36">tt1</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">(<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">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="c3">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="c37">tt2</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="c3">X</font>,<font color="Maroon" title="c2">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="c3">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="c37">tt2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T29" title="FOMODEL0:th.29">FOMODEL0:29</a></span>;<br/><a NAME="E17:181_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E112">CC2</font></span>: 
( <font color="Maroon" title="c16">I</font> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> (<span class="p1">(<span class="default">(<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span>,<font color="Maroon" title="c36">tt1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">EqClass</a> (<span class="p1">(<span class="default">(<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span>,<font color="Maroon" title="c37">tt2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:181_1_1"><span class="lab"><font color="Green" title="E109">C8</font></span></a>, <a class="txt" href="#E14:181_1_1"><span class="lab"><font color="Green" title="E111">C1</font></span></a>, <a class="txt" href="#E12:181"><span class="lab"><font color="Green" title="E104">B1</font></span></a>, <a class="txt" href="#E1:181_1_1"><span class="lab"><font color="Green" title="E106">C6</font></span></a>, <a class="txt" href="#E13:181"><span class="lab"><font color="Green" title="E105">B2</font></span></a>, <a class="ref" href="fomodel2.html#T15" title="FOMODEL2:th.15">FOMODEL2:15</a></span>;<br/><a NAME="E18:181_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">C2</font></span>: 
( <font color="Maroon" title="c16">I</font> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c36">tt1</font>,<font color="Maroon" title="c37">tt2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T35" title="EQREL_1:th.35">EQREL_1:35</a></span>;<br/><a NAME="E19:181_1_1"/><span class="lab"><font color="Green" title="E114">C7</font></span>: 
<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#K34" title="FOMODEL2:func.34">TheEqSymbOf</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> <font color="Maroon" title="c38">t1</font></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c39">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">phi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:181"><span class="lab"><font color="Green" title="E103">B5</font></span></a>, <a class="txt" href="#E5:181_1_1"><span class="lab"><font color="Green" title="E107">C3</font></span></a>, <a class="txt" href="#E6:181_1_1"><span class="lab"><font color="Green" title="E108">C4</font></span></a>, <a class="txt" href="#E12:181_1_1"><span class="lab"><font color="Green" title="E110">C5</font></span></a>, <a class="txt" href="#E1:181_1_1"><span class="lab"><font color="Green" title="E106">C6</font></span></a>, <a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></span>;<br/><span class="kw">thus </span><a NAME="E20:181_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="c4">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="181_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:181_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">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="c4">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font></span><br/>

<a NAME="E2:181_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c36">tt1</font>,<font color="Maroon" title="c37">tt2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">D1</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:181_1_1"><span class="lab"><font color="Green" title="E112">CC2</font></span></a>, <a class="ref" href="eqrel_1.html#T35" title="EQREL_1:th.35">EQREL_1:35</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c40">t11</font>, <font color="Maroon" title="c41">t22</font> being   <a href="fomodel1.html#V13" title="FOMODEL1:attr.13">termal</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E4:181_1_1_1"/><span class="lab"><font color="Green" title="E115">D0</font></span>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c36">tt1</font>,<font color="Maroon" title="c37">tt2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c40">t11</font>,<font color="Maroon" title="c41">t22</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</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> <font color="Maroon" title="c40">t11</font></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c41">t22</font> is <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">D1</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a>  )
 ;<br/>
<a NAME="E5:181_1_1_1"/>
( <font color="Maroon" title="c40">t11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">tt1</font> &amp; <font color="Maroon" title="c41">t22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">tt2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:181_1_1_1"><span class="lab"><font color="Green" title="E115">D0</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E6:181_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</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="p1">(<span class="default"><font color="Maroon" title="c40">t11</font> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c41">t22</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">phi</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:181"><span class="lab"><font color="Green" title="E103">B5</font></span></a>, <a class="txt" href="#E5:181_1_1"><span class="lab"><font color="Green" title="E107">C3</font></span></a>, <a class="txt" href="#E12:181_1_1"><span class="lab"><font color="Green" title="E110">C5</font></span></a>, <a class="txt" href="#E1:181_1_1"><span class="lab"><font color="Green" title="E106">C6</font></span></a>, <a class="ref" href="fomodel0.html#T15" title="FOMODEL0:th.15">FOMODEL0:15</a></span>;<br/>
<a NAME="E7:181_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">phi</font> is <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">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="#E4:181_1_1_1"><span class="lab"><font color="Green" title="E115">D0</font></span></a>, <a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></span>;<br/>
<a NAME="E8:181_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">phi</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="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D16" target="_self" title="FOMODEL4:def.16">DefExpanded</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:181_1_1_1"/>
<font color="Maroon" title="c4">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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>; <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="E21:181_1_1"/>
<font color="Maroon" title="c4">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 ; <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="c2">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c40">Xphi</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">phi</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">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="E23:181_1_1"/>
<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="c4">phi</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">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> is  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<font color="Maroon" title="c2">D1</font> <a href="fomodel4.html#V3" title="FOMODEL4:attr.3">-derivable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E35"><span class="lab"><font color="Green" title="E30">Lm20</font></span></a>, <a class="txt" href="#E2:181"><span class="lab"><font color="Green" title="E100">B10</font></span></a></span>;<br/><a NAME="E24:181_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">phi</font> is <font color="Maroon" title="c40">Xphi</font>,<font color="Maroon" title="c2">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#D12" target="_self" title="FOMODEL4:def.12">DefProvable</a></span>;<br/><span class="kw">hence </span><a NAME="E25:181_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">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="#E18:181_1_1"><span class="lab"><font color="Green" title="E113">C2</font></span></a>, <a class="txt" href="#E19:181_1_1"><span class="lab"><font color="Green" title="E114">C7</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="181_1_2"><span class="kw">suppose </span></a><a NAME="E1:181_1_2"/><span class="lab"><font color="Green" title="E106">C2</font></span>: 
 not <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="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <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"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">o</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="c4">phi</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#T15" title="FOMODEL1:th.15">FOMODEL1:15</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c33">gg</font> = <span class="p1">(<span class="default">(<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">X</font>) <a href="fomodel3.html#K7" title="FOMODEL3:func.7">-freeInterpreter</a></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c32">o</font>;<br/><a NAME="E3:181_1_2"/>
( <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="c4">phi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c16">I</font> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">X</font>) <a href="fomodel3.html#K7" title="FOMODEL3:func.7">-freeInterpreter</a></span>)</span> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">X</font>) <a href="fomodel3.html#K7" title="FOMODEL3:func.7">-freeInterpreter</a></span>)</span> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</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="c1">S</font>,<font color="Maroon" title="c3">X</font>) <a href="fomodel3.html#K7" title="FOMODEL3:func.7">-freeInterpreter</a></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c32">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c3">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_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K34" title="FOMODEL1:func.34">SubTerms</a> <font color="Maroon" title="c4">phi</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:181_1_2"><span class="lab"><font color="Green" title="E106">C2</font></span></a>, <a class="ref" href="fomodel2.html#T14" title="FOMODEL2:th.14">FOMODEL2:14</a>, <a class="ref" href="fomodel3.html#T5" title="FOMODEL3:th.5">FOMODEL3:5</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c16">I</font> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">X</font>) <a href="fomodel3.html#K7" title="FOMODEL3:func.7">-freeInterpreter</a></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c32">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c29">fp</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel3.html#T4" title="FOMODEL3:th.4">FOMODEL3:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">(<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">X</font>) <a href="fomodel3.html#K7" title="FOMODEL3:func.7">-freeInterpreter</a></span>)</span> <a href="fomodel2.html#K4" title="FOMODEL2:func.4">.</a> <font color="Maroon" title="c32">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">fp</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="fomodel3.html#K4" title="FOMODEL3:func.4">-freeInterpreter</a> <font color="Maroon" title="c32">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K34" title="FOMODEL1:func.34">SubTerms</a> <font color="Maroon" title="c4">phi</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel3.html#D4" title="FOMODEL3:def.4">FOMODEL3:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel0.html#K9" title="FOMODEL0:func.9">chi</a> (<font color="Maroon" title="c3">X</font>,<span class="p3">(<span class="default"><a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c1">S</font></span>)</span>)</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="c32">o</font> <a href="fomodel3.html#K2" title="FOMODEL3:func.2">-compound</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p5">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p0">(<span class="default"><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="c4">phi</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p4">(<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></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">pp</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel3.html#D3" title="FOMODEL3:def.3">FOMODEL3:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fomodel0.html#K9" title="FOMODEL0:func.9">chi</a> (<font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">nc</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">pp</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fomodel0.html#K9" title="FOMODEL0:func.9">chi</a> (<font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c32">o</font> <a href="fomodel3.html#K2" title="FOMODEL3:func.2">-compound</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">pp</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fomodel0.html#K9" title="FOMODEL0:func.9">chi</a> (<font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">o</font> <a href="fomodel3.html#K1" title="FOMODEL3:func.1">-compound</a> <font color="Maroon" title="c30">Pp</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel3.html#D2" title="FOMODEL3:def.2">FOMODEL3:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fomodel0.html#K9" title="FOMODEL0:func.9">chi</a> (<font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">phi</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#D38" title="FOMODEL1:def.38">FOMODEL1:def 38</a></span>
;<br/><a NAME="E5:181_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c16">I</font> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fomodel0.html#K9" title="FOMODEL0:func.9">chi</a> (<font color="Maroon" title="c3">X</font>,<span class="p2">(<span class="default"><a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c1">S</font></span>)</span>)</span>)</span> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T25" title="FOMODEL0:th.25">FOMODEL0:25</a></span>;<br/><a NAME="E6:181_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c16">I</font> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c1">S</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T24" title="FOMODEL0:th.24">FOMODEL0:24</a></span>;<br/><a NAME="E7:181_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c1">S</font> &amp; ( <font color="Maroon" title="c16">I</font> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies ( <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c1">S</font> ) ) &amp; ( <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c1">S</font> implies <font color="Maroon" title="c16">I</font> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:181_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="fomodel4.html#K39" title="FOMODEL4:func.39">Henkin</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff <font color="Maroon" title="c4">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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></div><span class="kw">end;</span></div>

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