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<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">  <a href="rewrite2.html#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <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="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <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><br/>

<a NAME="E1:48"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 <a href="rewrite2.html#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <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="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <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> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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><br/>

<span class="kw">assume </span><a NAME="E1:48_1"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <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"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:48_1"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  )
 <span class="kw">and </span><br/><a NAME="E4:48_1"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c4">b</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="#E3:48_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/>
<a NAME="E6:48_1"/>
<font color="Maroon" title="c5">a</font> <a href="rewrite2.html#R2" title="REWRITE2:pred.2">==&gt;.</a> <font color="Maroon" title="c6">b</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="#E1:48_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E4:48_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="rewrite2.html#D6" target="_self" title="REWRITE2:def.6">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:48_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="ref" href="rewrite2.html#T20" target="_self" title="REWRITE2:th.20">Th20</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="E2:48"/>
 <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oposet_1.html#D1" title="OPOSET_1:def.1">OPOSET_1:def 1</a></span>;<br/>
<a NAME="E3:48"/><span class="kw">then </span>
 <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> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="rewrite2.html#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <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 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">hence </span><a NAME="E4:48"/>
 <a href="rewrite2.html#K8" title="REWRITE2:func.8">==&gt;.-relation</a> <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="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <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="#E1:48"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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>
