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<b>let </b><font color="Maroon" title="c1">r</font>, <font color="Maroon" title="c2">s</font>, <font color="Maroon" title="c3">p</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="E1:80"/><i><font color="Green" title="E46">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">s</font>
 <b>and </b><br/><a NAME="E2:80"/><i><font color="Green" title="E47">A2</font></i>: 
<span class="p1"><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">].</a><span class="default"><font color="Maroon" title="c1">r</font>,<font color="Maroon" title="c2">s</font></span><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 ;<br/>

<a NAME="E3:80"/><i><font color="Green" title="E48">A3</font></i>: 
not <font color="Maroon" title="c1">r</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">r</font>,<font color="Maroon" title="c2">s</font></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" target="_self" title="XXREAL_1:th.4">Th4</a></i>;<br/>
<a NAME="E4:80"/><i><font color="Green" title="E49">A4</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font>
 
<b>by </b><i><a class="txt" href="xxreal_1.html#E1:80"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="xxreal_1.html#E2:80"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="xxreal_1.html#T51" target="_self" title="XXREAL_1:th.51">Th51</a></i>;<br/>
<a NAME="E5:80"/>
<font color="Maroon" title="c2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="txt" href="xxreal_1.html#E2:80"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="xxreal_1.html#E1:80"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="xxreal_1.html#T51" target="_self" title="XXREAL_1:th.51">Th51</a></i>;<br/>
<a NAME="E6:80"/><b>then </b>
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="txt" href="xxreal_1.html#E1:80"><i><font color="Green" title="E46">A1</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="E7:80"/>
contradiction
 <b>by </b><i><a class="txt" href="xxreal_1.html#E3:80"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="xxreal_1.html#E2:80"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="xxreal_1.html#E4:80"><i><font color="Green" title="E49">A4</font></i></a>, <a class="ref" href="xxreal_1.html#T1" target="_self" title="XXREAL_1:th.1">Th1</a></i>;<br/>


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