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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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">].</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#K4" title="XXREAL_1:func.4">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E1:211"/>
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#K4" title="XXREAL_1:func.4">].</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#K4" title="XXREAL_1:func.4">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="211_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="E2:211_1"/><i><font color="Green" title="E93">A1</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#K4" title="XXREAL_1:func.4">].</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#K4" title="XXREAL_1:func.4">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</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="#E2:211_1"><i><font color="Green" title="E93">A1</font></i></a></i>;<br/>
<a NAME="E4:211_1"/>
( <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  )
 
<b>by </b><i><a class="txt" href="#E2:211_1"><i><font color="Green" title="E93">A1</font></i></a>, <a class="ref" href="xxreal_1.html#T4" target="_self" title="XXREAL_1:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E5:211_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</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>hence </b><a NAME="E2:211"/>
<span class="p1"><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">].</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#K4" title="XXREAL_1:func.4">.[</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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