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<b>let </b><font color="Maroon" title="c2">r</font>, <font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">t</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="E1:235"/><i><font color="Green" title="E84">A1</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">s</font>
 <b>and </b><br/><a NAME="E2:235"/><i><font color="Green" title="E85">A2</font></i>: 
for <font color="Olive" title="b1">q</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>   st <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">q</font> &amp; <font color="Olive" title="b1">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">s</font> holds <br/><font color="Olive" title="b1">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t</font>
 ;<br/>

<a NAME="E3:235"/>
for <font color="Olive" title="b1">q</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>   st <font color="Maroon" title="c4">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">q</font> holds <br/><font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="235_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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="E1:235_1"/><i><font color="Green" title="E86">A3</font></i>: 
<font color="Maroon" title="c4">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">q</font>
 <b>and </b><br/><a NAME="E2:235_1"/><i><font color="Green" title="E87">A4</font></i>: 
<font color="Maroon" title="c5">q</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="235_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:235_1_1"/>
( <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">q</font> or <font color="Maroon" title="c5">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="235_1_1_1"><b>suppose </b></a><a NAME="E1:235_1_1_1"/>
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">q</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:235_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:235_1"><i><font color="Green" title="E86">A3</font></i></a>, <a class="txt" href="xreal_1.html#E2:235"><i><font color="Green" title="E85">A2</font></i></a>, <a class="txt" href="xreal_1.html#E2:235_1"><i><font color="Green" title="E87">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="235_1_1_2"><b>suppose </b></a><a NAME="E1:235_1_1_2"/><i><font color="Green" title="E88">A5</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c6">p</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="E3:235_1_1_2"/><i><font color="Green" title="E89">A6</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font>
 <b>and </b><br/><a NAME="E4:235_1_1_2"/><i><font color="Green" title="E90">A7</font></i>: 
<font color="Maroon" title="c6">p</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="xreal_1.html#E1:235"><i><font color="Green" title="E84">A1</font></i></a>, <a class="ref" href="xreal_1.html#T229" target="_self" title="XREAL_1:th.229">Th229</a></i>;<br/><a NAME="E5:235_1_1_2"/>
<font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E3:235_1_1_2"><i><font color="Green" title="E89">A6</font></i></a>, <a class="txt" href="xreal_1.html#E4:235_1_1_2"><i><font color="Green" title="E90">A7</font></i></a>, <a class="txt" href="xreal_1.html#E2:235"><i><font color="Green" title="E85">A2</font></i></a></i>;<br/><a NAME="E6:235_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E3:235_1_1_2"><i><font color="Green" title="E89">A6</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="E7:235_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="xreal_1.html#E1:235_1_1_2"><i><font color="Green" title="E88">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="xreal_1.html#E1:235_1"><i><font color="Green" title="E86">A3</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:235"/>
<font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">t</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T229" target="_self" title="XREAL_1:th.229">Th229</a></i>;<br/>


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