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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">r</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="E1:181"/><i><font color="Green" title="E60">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">r</font>
 <b>and </b><br/><a NAME="E2:181"/><i><font color="Green" title="E61">A2</font></i>: 
<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>
 <b>and </b><br/><a NAME="E3:181"/><i><font color="Green" title="E62">A3</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="c4">q</font>
 <b>and </b><br/><a NAME="E4:181"/><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c3">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">q</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="181_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:181_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="c3">s</font> or <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">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="181_1_1"><b>suppose </b></a><a NAME="E1:181_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="c3">s</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">r</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">].</a><span class="default"><font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">.[</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> = 
<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>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">s</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#T171" target="_self" title="XXREAL_1:th.171">Th171</a>, <a class="txt" href="xxreal_1.html#E1:181"><i><font color="Green" title="E60">A1</font></i></a></i>
<br/>.= 
<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>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">.[</a></span>
<b>by </b><i><a class="txt" href="xxreal_1.html#E2:181"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="xxreal_1.html#E4:181"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="xxreal_1.html#T173" target="_self" title="XXREAL_1:th.173">Th173</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="181_1_2"><b>suppose </b></a><a NAME="E1:181_1_2"/><i><font color="Green" title="E64">A5</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">r</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">r</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">].</a><span class="default"><font color="Maroon" title="c2">r</font>,<font color="Maroon" title="c3">s</font></span><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">.[</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">.[</a></span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">].</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">r</font></span><a href="xxreal_1.html#K3" title="XXREAL_1:func.3">.]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="xxreal_1.html#K2" title="XXREAL_1:func.2">[.</a><span class="default"><font color="Maroon" title="c3">s</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#T28" target="_self" title="XXREAL_1:th.28">Th28</a></i>
<br/>.= 
<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>,<font color="Maroon" title="c4">q</font></span><a href="xxreal_1.html#K4" title="XXREAL_1:func.4">.[</a></span>
<b>by </b><i><a class="txt" href="xxreal_1.html#E1:181_1_2"><i><font color="Green" title="E64">A5</font></i></a>, <a class="ref" href="xxreal_1.html#T178" target="_self" title="XXREAL_1:th.178">Th178</a>, <a class="txt" href="xxreal_1.html#E2:181"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="xxreal_1.html#E3:181"><i><font color="Green" title="E62">A3</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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