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

<span class="kw">let </span><font color="Maroon" title="c1">E</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">w</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  holds   <a href="rewrite2.html#K9" title="REWRITE2:func.9">Lang</a> <font color="Olive" title="b1">w</font>,<span class="p1">(<span class="default"><a href="oposet_1.html#K1" title="OPOSET_1:func.1">{}</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="flang_1.html#K4" title="FLANG_1:func.4">{</a><span class="default"><font color="Olive" title="b1">w</font></span><a href="flang_1.html#K4" title="FLANG_1:func.4">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">w</font> be    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="rewrite2.html#K9" title="REWRITE2:func.9">Lang</a> <font color="Maroon" title="c2">w</font>,<span class="p1">(<span class="default"><a href="oposet_1.html#K1" title="OPOSET_1:func.1">{}</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="flang_1.html#K4" title="FLANG_1:func.4">{</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="flang_1.html#K4" title="FLANG_1:func.4">}</a></span></span><br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">w</font> implies  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">Lang</a> <font color="Maroon" title="c2">w</font>,<span class="p1">(<span class="default"><a href="oposet_1.html#K1" title="OPOSET_1:func.1">{}</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:71_1"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">w</font>
 <span class="kw">and </span><br/><a NAME="E2:71_1"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">Lang</a> <font color="Maroon" title="c2">w</font>,<span class="p1">(<span class="default"><a href="oposet_1.html#K1" title="OPOSET_1:func.1">{}</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c4">t</font> = <font color="Maroon" title="c3">x</font> as    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a></span>;<br/><a NAME="E4:71_1"/>
<font color="Maroon" title="c2">w</font> <a href="rewrite2.html#R3" title="REWRITE2:pred.3">==&gt;*</a> <font color="Maroon" title="c4">t</font>, <a href="oposet_1.html#K1" title="OPOSET_1:func.1">{}</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="rewrite2.html#T46" target="_self" title="REWRITE2:th.46">Th46</a></span>;<br/><span class="kw">hence </span><a NAME="E5:71_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_1"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="rewrite2.html#T42" target="_self" title="REWRITE2:th.42">Th42</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:71"/><span class="kw">then </span>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K9" title="REWRITE2:func.9">Lang</a> <font color="Maroon" title="c2">w</font>,<span class="p1">(<span class="default"><a href="oposet_1.html#K1" title="OPOSET_1:func.1">{}</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span></span>)</span> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">w</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite2.html#T47" target="_self" title="REWRITE2:th.47">Th47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:71"/>
 <a href="rewrite2.html#K9" title="REWRITE2:func.9">Lang</a> <font color="Maroon" title="c2">w</font>,<span class="p1">(<span class="default"><a href="oposet_1.html#K1" title="OPOSET_1:func.1">{}</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="flang_1.html#K4" title="FLANG_1:func.4">{</a><span class="default"><font color="Maroon" title="c2">w</font></span><a href="flang_1.html#K4" title="FLANG_1:func.4">}</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
