<?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">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#K22" title="FOMODEL4:func.22">R0</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#K4" title="SUBSET_1:func.4">\/</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#K25" title="FOMODEL4:func.25">R3a</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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">D</font> &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>,<font color="Olive" title="b2">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</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">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#K22" title="FOMODEL4:func.22">R0</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#K4" title="SUBSET_1:func.4">\/</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#K25" title="FOMODEL4:func.25">R3a</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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">D</font> &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>,<font color="Olive" title="b1">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">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#K22" title="FOMODEL4:func.22">R0</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#K4" title="SUBSET_1:func.4">\/</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#K25" title="FOMODEL4:func.25">R3a</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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V10" title="FOMODEL4:attr.10">-expanded</a>  implies (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">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="c5">E</font> =  <a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">Q</font> = <font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">AF</font> =  <a href="fomodel2.html#K33" title="FOMODEL2:func.33">AllFormulasOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">Phi</font> = <font color="Maroon" title="c1">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">R</font> = (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">Seqts</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:96"/><span class="lab"><font color="Green" title="E72">B7</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="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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#K25" title="FOMODEL4:func.25">R3a</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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V10" title="FOMODEL4:attr.10">-expanded</a>  or (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:96"/><span class="lab"><font color="Green" title="E73">B6</font></span>: 
<font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c3">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>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> </span><br/>

<a NAME="E4:96"/><span class="lab"><font color="Green" title="E74">B9</font></span>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></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="c2">S</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T8" title="RELSET_1:th.8">RELSET_1:8</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c13">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:96_1"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">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"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">tt1</font> = <font color="Maroon" title="c11">x</font>, <font color="Maroon" title="c15">tt2</font> = <font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c16">tt3</font> = <font color="Maroon" title="c13">z</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="fomodel1.html#K35" title="FOMODEL1:func.35">AllTermsOf</a> <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:96"><span class="lab"><font color="Green" title="E74">B9</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">t1</font> = <font color="Maroon" title="c14">tt1</font>, <font color="Maroon" title="c18">t2</font> = <font color="Maroon" title="c15">tt2</font>, <font color="Maroon" title="c19">t3</font> = <font color="Maroon" title="c16">tt3</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="c2">S</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">phi1</font> = <span class="p1">(<span class="default"><span class="p2"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <font color="Maroon" title="c2">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="c17">t1</font></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c18">t2</font> as   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E2"><span class="lab"><font color="Green" title="E2">Lm10</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">phi2</font> = <span class="p1">(<span class="default"><span class="p2"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <font color="Maroon" title="c2">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="c18">t2</font></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c19">t3</font> as   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E2"><span class="lab"><font color="Green" title="E2">Lm10</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">phi3</font> = <span class="p1">(<span class="default"><span class="p2"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <font color="Maroon" title="c2">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="c17">t1</font></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c19">t3</font> as   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E2"><span class="lab"><font color="Green" title="E2">Lm10</font></span></a></span>;<br/><a NAME="E7:96_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">phi1</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><span class="p0">(<span class="default"><a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <font color="Maroon" title="c2">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="c18">t2</font></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c19">t3</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><span class="p5">(<span class="default"><a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <font color="Maroon" title="c2">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="c17">t1</font></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c19">t3</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is 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="c20">phi1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&lt;*</a><span class="default"><span class="p0">(<span class="default"><a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <font color="Maroon" title="c2">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="c17">t1</font></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c18">t2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="lang1.html#K4" title="LANG1:func.4">}</a></span>,<span class="p1"><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#K25" title="FOMODEL4:func.25">R3a</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="fomodel4.html#V4" title="FOMODEL4:attr.4">-derivable</a> 
 ;<br/><a NAME="E8:96_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">B2</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">phi1</font>,<font color="Maroon" title="c21">phi2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c22">phi3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is 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="c20">phi1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c20">phi1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="lang1.html#K4" title="LANG1:func.4">}</a></span>,<span class="p1"><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#K25" title="FOMODEL4:func.25">R3a</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="fomodel4.html#V4" title="FOMODEL4:attr.4">-derivable</a> 
 ;<br/><a NAME="E9:96_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">phi1</font>,<font color="Maroon" title="c21">phi2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c22">phi3</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</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#K4" title="SUBSET_1:func.4">\/</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#K25" title="FOMODEL4:func.25">R3a</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="fomodel4.html#V4" title="FOMODEL4:attr.4">-derivable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:96_1"><span class="lab"><font color="Green" title="E75">B2</font></span></a>, <a class="txt" href="fomodel4.html#E38"><span class="lab"><font color="Green" title="E33">Lm28</font></span></a></span>;<br/><a NAME="E10:96_1"/><span class="kw">then </span>
<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#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">phi1</font>,<font color="Maroon" title="c21">phi2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c22">phi3</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> ,<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="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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#K25" title="FOMODEL4:func.25">R3a</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="fomodel4.html#V3" title="FOMODEL4:attr.3">-derivable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E29"><span class="lab"><font color="Green" title="E24">Lm29</font></span></a></span>;<br/><a NAME="E11:96_1"/><span class="kw">then </span>
<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#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">phi1</font>,<font color="Maroon" title="c21">phi2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c22">phi3</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="c3">D</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="#E2:96"><span class="lab"><font color="Green" title="E72">B7</font></span></a></span>;<br/><a NAME="E12:96_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">BB3</font></span>: 
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">phi1</font>,<font color="Maroon" title="c21">phi2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</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"><font color="Maroon" title="c20">phi1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">phi2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c22">phi3</font> is <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c20">phi1</font>,<font color="Maroon" title="c21">phi2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c3">D</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>, <a class="ref" href="enumset1.html#T1" title="ENUMSET1:th.1">ENUMSET1:1</a></span>;<br/><span class="kw">assume </span><a NAME="E13:96_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</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="c11">x</font>,<font color="Maroon" title="c13">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a> </span><br/><a NAME="E14:96_1"/><span class="kw">then </span>
( <font color="Maroon" title="c20">phi1</font> is <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a>  &amp; <font color="Maroon" title="c21">phi2</font> is <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E39"><span class="lab"><font color="Green" title="E34">Lm23</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">Phi1</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c20">phi1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, <font color="Maroon" title="c24">Phi2</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">phi2</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="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:96"><span class="lab"><font color="Green" title="E73">B6</font></span></a>, <a class="ref" href="fomodel4.html#D16" target="_self" title="FOMODEL4:def.16">DefExpanded</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c25">Phi3</font> = <font color="Maroon" title="c23">Phi1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c24">Phi2</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E17:96_1"/>
<font color="Maroon" title="c22">phi3</font> is <font color="Maroon" title="c25">Phi3</font>,<font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:96_1"><span class="lab"><font color="Green" title="E76">BB3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E18:96_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a> 
 ; <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="E6:96"/><span class="kw">then </span>
(<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:96"/>
(<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">D</font>) <a href="fomodel4.html#K6" title="FOMODEL4:func.6">-termEq</a>  is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</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>
