<?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  <a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Olive" title="b1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">D</font> &amp; <font color="Maroon" title="c1">X</font> is <font color="Olive" title="b1">S</font> <a href="fomodel2.html#V9" title="FOMODEL2:attr.9">-covering</a>  &amp;  not <font color="Maroon" title="c1">X</font> is <font color="Olive" title="b2">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  holds <br/>( <font color="Maroon" title="c1">X</font> is <font color="Olive" title="b1">S</font> <a href="fomodel2.html#V6" title="FOMODEL2:attr.6">-mincover</a>  &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>  )</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  <a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> &amp; <font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c2">S</font> <a href="fomodel2.html#V9" title="FOMODEL2:attr.9">-covering</a>  &amp;  not <font color="Maroon" title="c1">X</font> is <font color="Olive" title="b1">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  holds <br/>( <font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c2">S</font> <a href="fomodel2.html#V6" title="FOMODEL2:attr.6">-mincover</a>  &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>  )</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"> (  <a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c2">S</font> <a href="fomodel2.html#V9" title="FOMODEL2:attr.9">-covering</a>  &amp;  not <font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</a>  implies ( <font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c2">S</font> <a href="fomodel2.html#V6" title="FOMODEL2:attr.6">-mincover</a>  &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>  ) )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">G0</font> =  <a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">E0</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:193"/><span class="lab"><font color="Green" title="E108">B0</font></span>: 
 <a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font>
 <span class="kw">and </span><br/><a NAME="E2:193"/><span class="lab"><font color="Green" title="E109">B1</font></span>: 
( <font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c2">S</font> <a href="fomodel2.html#V9" title="FOMODEL2:attr.9">-covering</a>  &amp;  not <font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c3">D</font> <a href="fomodel4.html#V14" title="FOMODEL4:attr.14">-inconsistent</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> is <font color="Maroon" title="c2">S</font> <a href="fomodel2.html#V6" title="FOMODEL2:attr.6">-mincover</a>  &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>  )</span><br/>

<a NAME="E3:193"/><span class="lab"><font color="Green" title="E110">B2</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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:193"><span class="lab"><font color="Green" title="E108">B0</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<a NAME="E4:193"/><span class="lab"><font color="Green" title="E111">BBB3</font></span>: 
 for <font color="Olive" title="b1">phi</font> being   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font> holds <br/> ( ( <font color="Olive" title="b1">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies  not  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Olive" title="b1">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> ) &amp; (  not <font color="Olive" title="b1">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Olive" title="b1">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="193_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">phi</font> be   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c6">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies  not  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c6">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> ) &amp; (  not <font color="Maroon" title="c6">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c6">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> ) )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c6">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:193_1_1"/>
<font color="Maroon" title="c6">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c6">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><a NAME="E2:193_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">phi</font> is <font color="Maroon" title="c1">X</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#T4" target="_self" title="FOMODEL4:th.4">Lm2</a></span>;<br/><a NAME="E3:193_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">phi</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="#E3:193"><span class="lab"><font color="Green" title="E110">B2</font></span></a>, <a class="txt" href="fomodel4.html#E110"><span class="lab"><font color="Green" title="E89">Lm35</font></span></a></span>;<br/><a NAME="E4:193_1_1"/><span class="kw">then </span>
 not  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c6">phi</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="#E2:193"><span class="lab"><font color="Green" title="E109">B1</font></span></a>, <a class="ref" href="fomodel4.html#D65" target="_self" title="FOMODEL4:def.65">DefInc</a></span>;<br/><a NAME="E5:193_1_1"/><span class="kw">then </span>
 not  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c6">phi</font> is <font color="Maroon" title="c1">X</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:193"><span class="lab"><font color="Green" title="E110">B2</font></span></a>, <a class="txt" href="fomodel4.html#E110"><span class="lab"><font color="Green" title="E89">Lm35</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:193_1_1"/>
 not  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c6">phi</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="ref" href="fomodel4.html#T4" target="_self" title="FOMODEL4:th.4">Lm2</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="E2:193_1"/>
 not <font color="Maroon" title="c6">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c6">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/>

<span class="kw">hence </span><a NAME="E3:193_1"/>
 <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c6">phi</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="#E2:193"><span class="lab"><font color="Green" title="E109">B1</font></span></a>, <a class="ref" href="fomodel2.html#D40" title="FOMODEL2:def.40">FOMODEL2:def 40</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="E5:193"/><span class="kw">then </span>
 for <font color="Olive" title="b1">phi</font> being   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font> holds <br/> ( <font color="Olive" title="b1">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> iff  not  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Olive" title="b1">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E6:193"/>
<font color="Maroon" title="c1">X</font> is <font color="Maroon" title="c2">S</font> <a href="fomodel2.html#V6" title="FOMODEL2:attr.6">-mincover</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#D34" title="FOMODEL2:def.34">FOMODEL2:def 34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="193_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">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"> ( <font color="Maroon" title="c6">x</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>  implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )</span><br/><span class="kw">assume </span><a NAME="E1:193_2"/><span class="lab"><font color="Green" title="E112">C0</font></span>: 
<font color="Maroon" title="c6">x</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> 
 ; <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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">phi</font> = <font color="Maroon" title="c6">x</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#E41"><span class="lab"><font color="Green" title="E36">Lm33</font></span></a></span>;<br/><a NAME="E3:193_2"/>
 not  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c7">phi</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="#E1:193_2"><span class="lab"><font color="Green" title="E112">C0</font></span></a>, <a class="txt" href="#E2:193"><span class="lab"><font color="Green" title="E109">B1</font></span></a>, <a class="ref" href="fomodel4.html#D65" target="_self" title="FOMODEL4:def.65">DefInc</a></span>;<br/><a NAME="E4:193_2"/><span class="kw">then </span>
 not  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c7">phi</font> is <font color="Maroon" title="c1">X</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c2">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="fomodel4.html#V8" title="FOMODEL4:attr.8">-provable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:193"><span class="lab"><font color="Green" title="E110">B2</font></span></a>, <a class="txt" href="fomodel4.html#E110"><span class="lab"><font color="Green" title="E89">Lm35</font></span></a></span>;<br/><a NAME="E5:193_2"/><span class="kw">then </span>
 not  <a href="fomodel2.html#K38" title="FOMODEL2:func.38">xnot</a> <font color="Maroon" title="c7">phi</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="ref" href="fomodel4.html#T4" target="_self" title="FOMODEL4:th.4">Lm2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:193_2"/>
<font color="Maroon" title="c6">x</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="#E4:193"><span class="lab"><font color="Green" title="E111">BBB3</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>
<span class="kw">hence </span><a NAME="E8:193"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D61" target="_self" title="FOMODEL4:def.61">DefExpanded2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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