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<b>let </b><font color="Maroon" title="c1">x</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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="membered.html#D14" title="MEMBERED:def.14">MEMBERED:def 14</a><br/>

<b>thus </b><a NAME="E1:19"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  implies <font color="Maroon" title="c1">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"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+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="19_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ;<br/>

<a NAME="E2:19_1"/><b>then </b>
(  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1"/>
<font color="Maroon" title="c1">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"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">.[</a></span>
 <b>by </b><i><a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></i>;<br/>


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

<b>assume </b><a NAME="E2:19"/>
<font color="Maroon" title="c1">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"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">.[</a></span>
 ;<br/>

<a NAME="E3:19"/><b>then </b>
(  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="xxreal_1.html#T4" title="XXREAL_1:th.4">XXREAL_1:4</a></i>;<br/>
<b>hence </b><a NAME="E4:19"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/>


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