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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">  <a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c1">S</font> is  <a href="fomodel4.html#V17" title="FOMODEL4:attr.17">Correct</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="200_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c2">f</font> =  <a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c1">S</font>;<br/><span class="kw">set </span><font color="Maroon" title="c3">R</font> =  <a href="fomodel4.html#K8" title="FOMODEL4:func.8">P0</a> <font color="Maroon" title="c1">S</font>;<br/><span class="kw">set </span><font color="Maroon" title="c4">Q</font> = <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a> ;<br/><span class="kw">let </span><font color="Maroon" title="c5">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="c5">X</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel2.html#V12" title="FOMODEL2:attr.12">-correct</a>  implies <span class="p1">(<span class="default"><a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">X</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel2.html#V12" title="FOMODEL2:attr.12">-correct</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:200_1"/>
<font color="Maroon" title="c5">X</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel2.html#V12" title="FOMODEL2:attr.12">-correct</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">X</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel2.html#V12" title="FOMODEL2:attr.12">-correct</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="200_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">U</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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">I</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <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="fomodel2.html#V10" title="FOMODEL2:attr.10">-satisfied</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b3">phi</font> being   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">phi</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="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">X</font> holds <br/><font color="Olive" title="b1">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Olive" title="b3">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">set </span><font color="Maroon" title="c7">II</font> = <font color="Maroon" title="c6">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c1">S</font>;<br/><span class="kw">let </span><font color="Maroon" title="c8">I</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</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">  for <font color="Olive" title="b1">x</font> being  <font color="Maroon" title="c8">I</font> <a href="fomodel2.html#V10" title="FOMODEL2:attr.10">-satisfied</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/>  for <font color="Olive" title="b2">phi</font> being   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">phi</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="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">X</font> holds <br/><font color="Maroon" title="c8">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Olive" title="b2">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c9">x</font> be  <font color="Maroon" title="c8">I</font> <a href="fomodel2.html#V10" title="FOMODEL2:attr.10">-satisfied</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="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">phi</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="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">X</font> holds <br/><font color="Maroon" title="c8">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Olive" title="b1">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c10">phi</font> be   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">phi</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="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">X</font> implies <font color="Maroon" title="c8">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c10">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">set </span><font color="Maroon" title="c11">s</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">phi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/><a NAME="E1:200_1_1"/>
( <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">phi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <font color="Maroon" title="c9">x</font> <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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">phi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <font color="Maroon" title="c10">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/><a NAME="E2:200_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">C0</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="fomodel4.html#K8" title="FOMODEL4:func.8">P0</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">phi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">phi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">phi</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T29" title="FOMODEL0:th.29">FOMODEL0:29</a></span>;<br/><span class="kw">assume </span><a NAME="E3:200_1_1"/><span class="lab"><font color="Green" title="E114">DD1</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">phi</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="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="c8">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c10">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E4:200_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">D1</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">phi</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">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">X</font>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">phi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></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="fomodel4.html#K8" title="FOMODEL4:func.8">P0</a> <font color="Maroon" title="c1">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel4.html#E73"><span class="lab"><font color="Green" title="E68">Lm1e</font></span></a></span>;<br/><a NAME="E5:200_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="fomodel4.html#K8" title="FOMODEL4:func.8">P0</a> <font color="Maroon" title="c1">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">XX</font> = <font color="Maroon" title="c5">X</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a></span>)</span> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">seqt</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">phi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">S</font> <a href="fomodel4.html#K1" title="FOMODEL4:func.1">-sequents</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:200_1_1"><span class="lab"><font color="Green" title="E114">DD1</font></span></a>, <a class="txt" href="fomodel4.html#E73"><span class="lab"><font color="Green" title="E68">Lm1e</font></span></a></span>;<br/><a NAME="E8:200_1_1"/>
<font color="Maroon" title="c13">seqt</font> <a href="fomodel4.html#R1" title="FOMODEL4:pred.1">Rule0</a> <font color="Maroon" title="c12">XX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:200_1_1"><span class="lab"><font color="Green" title="E115">D1</font></span></a>, <a class="ref" href="fomodel4.html#D31" target="_self" title="FOMODEL4:def.31">DefP0</a></span>;<br/><a NAME="E9:200_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">phi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:200_1_1"><span class="lab"><font color="Green" title="E113">C0</font></span></a>, <a class="ref" href="fomodel4.html#D18" target="_self" title="FOMODEL4:def.18">Def0</a></span>;<br/><span class="kw">hence </span><a NAME="E10:200_1_1"/>
<font color="Maroon" title="c8">I</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c10">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="fomodel2.html#D42" title="FOMODEL2:def.42">FOMODEL2:def 42</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="E3:200_1"/>
<span class="p1">(<span class="default"><a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">X</font> is <font color="Maroon" title="c1">S</font> <a href="fomodel2.html#V12" title="FOMODEL2:attr.12">-correct</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#D44" title="FOMODEL2:def.44">FOMODEL2:def 44</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="E2:200"/>
 <a href="fomodel4.html#K22" title="FOMODEL4:func.22">R0</a> <font color="Maroon" title="c1">S</font> is  <a href="fomodel4.html#V17" title="FOMODEL4:attr.17">Correct</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel4.html#D69" target="_self" title="FOMODEL4:def.69">RuleCorr</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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