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

<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="fomodel1.html#K8" title="FOMODEL1:func.8">TheNorSymbOf</a> <font color="Maroon" title="c1">S</font> &amp; <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="fomodel1.html#K18" title="FOMODEL1:func.18">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font> holds <br/><font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">s</font> be   <a href="fomodel1.html#NM1" title="FOMODEL1:NM.1">Element</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"> ( <font color="Maroon" title="c2">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="fomodel1.html#K8" title="FOMODEL1:func.8">TheNorSymbOf</a> <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c2">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="fomodel1.html#K18" title="FOMODEL1:func.18">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font> implies <font color="Maroon" title="c2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">O</font> =  <a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">R</font> =  <a href="fomodel1.html#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">E</font> =  <a href="fomodel1.html#K18" title="FOMODEL1:func.18">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">X</font> = <span class="p1">(<span class="default"><a href="fomodel1.html#K4" title="FOMODEL1:func.4">RelSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">N</font> =  <a href="fomodel1.html#K8" title="FOMODEL1:func.8">TheNorSymbOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">SS</font> =  <a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">assume </span><a NAME="E1:101"/>
( <font color="Maroon" title="c2">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="fomodel1.html#K8" title="FOMODEL1:func.8">TheNorSymbOf</a> <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c2">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="fomodel1.html#K18" title="FOMODEL1:func.18">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font></span><br/>

<a NAME="E2:101"/><span class="kw">then </span>
(  not <font color="Maroon" title="c2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K8" title="FOMODEL1:func.8">TheNorSymbOf</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  not <font color="Maroon" title="c2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="fomodel1.html#K18" title="FOMODEL1:func.18">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E3:101"/><span class="kw">then </span>
 not <font color="Maroon" title="c2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K8" title="FOMODEL1:func.8">TheNorSymbOf</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</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="fomodel1.html#K18" title="FOMODEL1:func.18">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E4:101"/><span class="kw">then </span>
 not <font color="Maroon" title="c2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K8" title="FOMODEL1:func.8">TheNorSymbOf</a> <font color="Maroon" title="c1">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="fomodel1.html#K18" title="FOMODEL1:func.18">TheEqSymbOf</a> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T1" title="ENUMSET1:th.1">ENUMSET1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:101"/>
<font color="Maroon" title="c2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</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>
