<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a><br/>  for <font color="Olive" title="b2">r</font> being   <a href="fomodel1.html#V7" title="FOMODEL1:attr.7">relational</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">D</font> being   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Olive" title="b1">S</font>  st <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="Olive" title="b1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.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#K24" title="FOMODEL4:func.24">R2</a> <font color="Olive" title="b1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &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"><a href="fomodel4.html#K26" title="FOMODEL4:func.26">R3b</a> <font color="Olive" title="b1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b3">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.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#K26" title="FOMODEL4:func.26">R3b</a> <font color="Olive" title="b1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b3">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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#K28" title="FOMODEL4:func.28">R3e</a> <font color="Olive" title="b1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.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#K28" title="FOMODEL4:func.28">R3e</a> <font color="Olive" title="b1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c1">X</font> is <font color="Olive" title="b3">D</font> <a href="fomodel4.html#V10" title="FOMODEL4:attr.10">-expanded</a>  holds <br/><font color="Maroon" title="c1">X</font> <a href="fomodel3.html#K4" title="FOMODEL3:func.4">-freeInterpreter</a> <font color="Olive" title="b2">r</font> is (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  <a href="fomodel3.html#V2" title="FOMODEL3:attr.2">-respecting</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">r</font> being   <a href="fomodel1.html#V7" title="FOMODEL1:attr.7">relational</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">D</font> being   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Maroon" title="c2">S</font>  st <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="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.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#K24" title="FOMODEL4:func.24">R2</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &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"><a href="fomodel4.html#K26" title="FOMODEL4:func.26">R3b</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.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#K26" title="FOMODEL4:func.26">R3b</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b2">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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#K28" title="FOMODEL4:func.28">R3e</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.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#K28" title="FOMODEL4:func.28">R3e</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c1">X</font> is <font color="Olive" title="b2">D</font> <a href="fomodel4.html#V10" title="FOMODEL4:attr.10">-expanded</a>  holds <br/><font color="Maroon" title="c1">X</font> <a href="fomodel3.html#K4" title="FOMODEL3:func.4">-freeInterpreter</a> <font color="Olive" title="b1">r</font> is (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  <a href="fomodel3.html#V2" title="FOMODEL3:attr.2">-respecting</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="fomodel1.html#V7" title="FOMODEL1:attr.7">relational</a>  <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">D</font> being   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Maroon" title="c2">S</font>  st <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="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.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#K24" title="FOMODEL4:func.24">R2</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &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"><a href="fomodel4.html#K26" title="FOMODEL4:func.26">R3b</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.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#K26" title="FOMODEL4:func.26">R3b</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Olive" title="b1">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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#K28" title="FOMODEL4:func.28">R3e</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.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#K28" title="FOMODEL4:func.28">R3e</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c1">X</font> is <font color="Olive" title="b1">D</font> <a href="fomodel4.html#V10" title="FOMODEL4:attr.10">-expanded</a>  holds <br/><font color="Maroon" title="c1">X</font> <a href="fomodel3.html#K4" title="FOMODEL3:func.4">-freeInterpreter</a> <font color="Maroon" title="c3">r</font> is (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  <a href="fomodel3.html#V2" title="FOMODEL3:attr.2">-respecting</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">D</font> be   <a href="fomodel4.html#NM2" title="FOMODEL4:NM.2">RuleSet</a> of <font color="Maroon" title="c2">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"><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="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.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#K24" title="FOMODEL4:func.24">R2</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &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"><a href="fomodel4.html#K26" title="FOMODEL4:func.26">R3b</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.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#K26" title="FOMODEL4:func.26">R3b</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c4">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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#K28" title="FOMODEL4:func.28">R3e</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.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#K28" title="FOMODEL4:func.28">R3e</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c4">D</font> <a href="fomodel4.html#V10" title="FOMODEL4:attr.10">-expanded</a>  implies <font color="Maroon" title="c1">X</font> <a href="fomodel3.html#K4" title="FOMODEL3:func.4">-freeInterpreter</a> <font color="Maroon" title="c3">r</font> is (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  <a href="fomodel3.html#V2" title="FOMODEL3:attr.2">-respecting</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:110"/><span class="lab"><font color="Green" title="E81">B0</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#K24" title="FOMODEL4:func.24">R2</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.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#K24" title="FOMODEL4:func.24">R2</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &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"><a href="fomodel4.html#K26" title="FOMODEL4:func.26">R3b</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.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#K26" title="FOMODEL4:func.26">R3b</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c4">D</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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#K28" title="FOMODEL4:func.28">R3e</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.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#K28" title="FOMODEL4:func.28">R3e</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c4">D</font> <a href="fomodel4.html#V10" title="FOMODEL4:attr.10">-expanded</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="c1">X</font> <a href="fomodel3.html#K4" title="FOMODEL3:func.4">-freeInterpreter</a> <font color="Maroon" title="c3">r</font> is (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  <a href="fomodel3.html#V2" title="FOMODEL3:attr.2">-respecting</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">AT</font> =  <a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">R</font> = (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">I</font> = <font color="Maroon" title="c1">X</font> <a href="fomodel3.html#K4" title="FOMODEL3:func.4">-freeInterpreter</a> <font color="Maroon" title="c3">r</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">AF</font> =  <a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">ch</font> =  <a href="fomodel0.html#K9" title="FOMODEL0:func.9">chi</a> (<font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c2">S</font></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c10">SS</font> =  <a href="fomodel1.html#K1" title="FOMODEL1:func.1">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c3">r</font> <a href="fomodel3.html#K2" title="FOMODEL3:func.2">-compound</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">m</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> <font color="Maroon" title="c3">r</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">x1</font>, <font color="Maroon" title="c14">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="fomodel3.html#K9" title="FOMODEL3:func.9">-placesOf</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="fomodel3.html#K4" title="FOMODEL3:func.4">-freeInterpreter</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="fomodel3.html#K4" title="FOMODEL3:func.4">-freeInterpreter</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel2.html#K3" title="FOMODEL2:func.3">id</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> )</span><br/><span class="kw">assume </span><a NAME="E1:110_1"/><span class="lab"><font color="Green" title="E82">C0</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="fomodel3.html#K9" title="FOMODEL3:func.9">-placesOf</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="fomodel3.html#K4" title="FOMODEL3:func.4">-freeInterpreter</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="fomodel3.html#K4" title="FOMODEL3:func.4">-freeInterpreter</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel2.html#K3" title="FOMODEL2:func.3">id</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">T1</font>, <font color="Maroon" title="c16">T2</font> being   <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> <font color="Maroon" title="c3">r</font></span>)</span> <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="c2">S</font></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a> <span class="kw"> such that </span><br/><a NAME="E3:110_1"/><span class="lab"><font color="Green" title="E83">C1</font></span>: 
( <font color="Maroon" title="c15">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x1</font> &amp; <font color="Maroon" title="c16">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x2</font> &amp;  <a href="fomodel4.html#K36" title="FOMODEL4:func.36">PairWiseEq</a> (<font color="Maroon" title="c15">T1</font>,<font color="Maroon" title="c16">T2</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E94"><span class="lab"><font color="Green" title="E75">Lm42</font></span></a>, <a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E81">B0</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">w1</font> = <font color="Maroon" title="c3">r</font> <a href="fomodel3.html#K1" title="FOMODEL3:func.1">-compound</a> <font color="Maroon" title="c15">T1</font>, <font color="Maroon" title="c18">w2</font> = <font color="Maroon" title="c3">r</font> <a href="fomodel3.html#K1" title="FOMODEL3:func.1">-compound</a> <font color="Maroon" title="c16">T2</font> as   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c2">S</font> ;<br/><a NAME="E5:110_1"/>
( <font color="Maroon" title="c17">w1</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="c2">S</font> &amp; <font color="Maroon" title="c18">w2</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="c2">S</font> )
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">w11</font> = <font color="Maroon" title="c17">w1</font>, <font color="Maroon" title="c20">w22</font> = <font color="Maroon" title="c18">w2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c2">S</font> ;<br/><a NAME="E7:110_1"/><span class="lab"><font color="Green" title="E84">C2</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</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="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">w11</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</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="c14">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">w22</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1"><span class="lab"><font color="Green" title="E83">C1</font></span></a>, <a class="ref" href="fomodel3.html#D2" title="FOMODEL3:def.2">FOMODEL3:def 2</a></span>;<br/><a NAME="E8:110_1"/>
( <font color="Maroon" title="c19">w11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> iff <font color="Maroon" title="c20">w22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:110_1"><span class="lab"><font color="Green" title="E84">C2</font></span></a>, <a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E81">B0</font></span></a>, <a class="txt" href="#E1:110_1"><span class="lab"><font color="Green" title="E82">C0</font></span></a>, <a class="txt" href="fomodel4.html#E97"><span class="lab"><font color="Green" title="E78">Lm46</font></span></a></span>;<br/><a NAME="E9:110_1"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel0.html#K9" title="FOMODEL0:func.9">chi</a> (<font color="Maroon" title="c1">X</font>,<span class="p4">(<span class="default"><a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c2">S</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">w1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel0.html#K9" title="FOMODEL0:func.9">chi</a> (<font color="Maroon" title="c1">X</font>,<span class="p4">(<span class="default"><a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c2">S</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">w2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel2.html#K3" title="FOMODEL2:func.3">id</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="fomodel3.html#K4" title="FOMODEL3:func.4">-freeInterpreter</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fomodel0.html#K9" title="FOMODEL0:func.9">chi</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c2">S</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">w1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="fomodel3.html#K4" title="FOMODEL3:func.4">-freeInterpreter</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fomodel0.html#K9" title="FOMODEL0:func.9">chi</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="fomodel1.html#K32" title="FOMODEL1:func.32">AtomicFormulasOf</a> <font color="Maroon" title="c2">S</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">w2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E98"><span class="lab"><font color="Green" title="E79">Lm47</font></span></a>, <a class="ref" href="fomodel3.html#T6" title="FOMODEL3:th.6">FOMODEL3:6</a></span>;<br/><span class="kw">hence </span><a NAME="E10:110_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="fomodel3.html#K4" title="FOMODEL3:func.4">-freeInterpreter</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="fomodel3.html#K4" title="FOMODEL3:func.4">-freeInterpreter</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel2.html#K3" title="FOMODEL2:func.3">id</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1"><span class="lab"><font color="Green" title="E83">C1</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>
<a NAME="E3:110"/><span class="kw">then </span>
<font color="Maroon" title="c1">X</font> <a href="fomodel3.html#K4" title="FOMODEL3:func.4">-freeInterpreter</a> <font color="Maroon" title="c3">r</font> is <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> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="fomodel3.html#K9" title="FOMODEL3:func.9">-placesOf</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span>, <a href="fomodel2.html#K3" title="FOMODEL2:func.3">id</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <a href="fomodel3.html#V1" title="FOMODEL3:attr.1">-respecting</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel3.html#D9" title="FOMODEL3:def.9">FOMODEL3:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:110"/>
<font color="Maroon" title="c1">X</font> <a href="fomodel3.html#K4" title="FOMODEL3:func.4">-freeInterpreter</a> <font color="Maroon" title="c3">r</font> is (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  <a href="fomodel3.html#V2" title="FOMODEL3:attr.2">-respecting</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel3.html#D10" title="FOMODEL3:def.10">FOMODEL3:def 10</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>
