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<b>let </b><font color="Maroon" title="c1">p</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:214"/><i><font color="Green" title="E62">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 ;<br/>

<a NAME="E2:214"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> </span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="214_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:214_1"/><i><font color="Green" title="E63">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> </span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c3">s</font> = <font color="Maroon" title="c2">x</font> as  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="txt" href="xxreal_1.html#E2:214_1"><i><font color="Green" title="E63">A2</font></i></a>, <a class="ref" href="membered.html#D2" title="MEMBERED:def.2">MEMBERED:def 2</a></i>;<br/>
<a NAME="E4:214_1"/>
( <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  )
 
<b>by </b><i><a class="txt" href="xxreal_1.html#E2:214_1"><i><font color="Green" title="E63">A2</font></i></a>, <a class="ref" href="xxreal_1.html#T1" target="_self" title="XXREAL_1:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E5:214_1"/>
contradiction
 <b>by </b><i><a class="txt" href="xxreal_1.html#E1:214"><i><font color="Green" title="E62">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="xxreal_0.html#T6" title="XXREAL_0:th.6">XXREAL_0:6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:214"/>
<span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> </span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>


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