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<b>let </b><font color="Maroon" title="c1">r</font>, <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">q</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><b>that </b><br/><a NAME="E1:104"/><i><font color="Green" title="E47">A1</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font>
 <b>and </b><br/><a NAME="E2:104"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">s</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:104_1"/>
( <font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font> or <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_1"><b>suppose </b></a><a NAME="E1:104_1_1"/><i><font color="Green" title="E49">A3</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c5">t</font> being  <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> such that </b><br/><a NAME="E3:104_1_1"/><i><font color="Green" title="E50">A4</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">t</font>
 <b>and </b><br/><a NAME="E4:104_1_1"/><i><font color="Green" title="E51">A5</font></i>: 
<font color="Maroon" title="c5">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="txt" href="xxreal_1.html#E2:104"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="xreal_1.html#T229" title="XREAL_1:th.229">XREAL_1:229</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">t</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="membered.html#D8" title="MEMBERED:def.8">MEMBERED:def 8</a><br/><b>thus </b><a NAME="E5:104_1_1"/>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="xxreal_1.html#E4:104_1_1"><i><font color="Green" title="E51">A5</font></i></a>, <a class="txt" href="xxreal_1.html#E3:104_1_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="xxreal_1.html#T3" target="_self" title="XXREAL_1:th.3">Th3</a></i>;<br/><a NAME="E6:104_1_1"/>
<font color="Maroon" title="c5">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="xxreal_1.html#E1:104_1_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="xxreal_1.html#E4:104_1_1"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E7:104_1_1"/>
not <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <b>by </b><i><a class="ref" href="xxreal_1.html#T3" target="_self" title="XXREAL_1:th.3">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_1_2"><b>suppose </b></a><a NAME="E1:104_1_2"/><i><font color="Green" title="E49">A6</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">s</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c5">t</font> being  <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> such that </b><br/><a NAME="E3:104_1_2"/><i><font color="Green" title="E50">A7</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">t</font>
 <b>and </b><br/><a NAME="E4:104_1_2"/><i><font color="Green" title="E51">A8</font></i>: 
<font color="Maroon" title="c5">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="xxreal_1.html#E1:104"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="xreal_1.html#T229" title="XREAL_1:th.229">XREAL_1:229</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">t</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="membered.html#D8" title="MEMBERED:def.8">MEMBERED:def 8</a><br/><a NAME="E5:104_1_2"/>
<font color="Maroon" title="c5">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">s</font>
 <b>by </b><i><a class="txt" href="xxreal_1.html#E1:104_1_2"><i><font color="Green" title="E49">A6</font></i></a>, <a class="txt" href="xxreal_1.html#E4:104_1_2"><i><font color="Green" title="E51">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E6:104_1_2"/>
<font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="xxreal_1.html#E3:104_1_2"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="xxreal_1.html#T3" target="_self" title="XXREAL_1:th.3">Th3</a></i>;<br/><b>thus </b><a NAME="E7:104_1_2"/>
not <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="xxreal_1.html#E4:104_1_2"><i><font color="Green" title="E51">A8</font></i></a>, <a class="ref" href="xxreal_1.html#T3" target="_self" title="XXREAL_1:th.3">Th3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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