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

<a NAME="E1:12_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> </span>}</span>  is <a href="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">e</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="membered.html#D2" title="MEMBERED:def.2">MEMBERED:def 2</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="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> </span>}</span>  or <font color="Maroon" title="c2">e</font> is <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a> )</span><br/>

<b>assume </b><a NAME="E1:12_1_1_1"/>
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">e</font> is <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a></span><br/>

<a NAME="E2:12_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">w</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  st <br/>( <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Olive" title="b1">w</font> &amp; <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> )
 
;<br/>
<b>hence </b><a NAME="E3:12_1_1_1"/>
<font color="Maroon" title="c2">e</font> is <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>
 ; <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>hence </b><a NAME="E2:12_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> </span>}</span>  is  <a href="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a>  <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"> verum</span><br/>


</div>
