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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">A</font> being  <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">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span>  st  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> holds <br/>( <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Olive" title="b2">A</font> )</span><br/><b>let </b><font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">A</font> be   <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">E</font> <a href="catalan2.html#K3" title="CATALAN2:func.3">^omega</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> implies ( <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">A</font> ) )</span><br/>





<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E17">A1</font></i>: 
 <a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">A</font> )</span><br/>

<b>thus </b><a NAME="E2:23"/>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c2">B</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">A</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c2">B</font> )</span><br/>

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c2">B</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">xa</font> = <font color="Maroon" title="c4">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#K3" title="CATALAN2:func.3">^omega</a>  ;<br/>
<a NAME="E3:23_1"/>
<font color="Maroon" title="c5">xa</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <span class="p1">(<span class="default"><a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="#E1:23"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E1:23_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="flang_1.html#D1" target="_self" title="FLANG_1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T32" title="AFINSQ_1:th.32">AFINSQ_1:32</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:23"/>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">A</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">A</font> )</span><br/>

<b>assume </b><a NAME="E1:23_2"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">A</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">xa</font> = <font color="Maroon" title="c4">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#K3" title="CATALAN2:func.3">^omega</a>  ;<br/>
<a NAME="E3:23_2"/>
<span class="p1">(<span class="default"><a href="flang_1.html#K2" title="FLANG_1:func.2">&lt;%&gt;</a> <font color="Maroon" title="c1">E</font></span>)</span> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c5">xa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="#E1:23"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E1:23_2"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="flang_1.html#D1" target="_self" title="FLANG_1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:23_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="flang_1.html#K6" title="FLANG_1:func.6">^^</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T32" title="AFINSQ_1:th.32">AFINSQ_1:32</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


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