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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>



<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E29">A1</font></i>: 
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> or  not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1"><b>suppose </b></a><a NAME="E1:46_1_1"/><i><font color="Green" title="E30">A2</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 ;<br/><div class="add"><a NAME="E2:46_1_1"/><b>then </b>
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="txt" href="extreal2.html#E1:46"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="extreal1.html#D3" title="EXTREAL1:def.3">EXTREAL1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:46_1_1"/>
(  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c2">y</font> <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> <font color="Maroon" title="c2">y</font> )
 <b>by </b><i><a class="txt" href="extreal2.html#E1:46_1_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="extreal2.html#T47" target="_self" title="EXTREAL2:th.47">Th47</a>, <a class="ref" href="extreal1.html#T24" title="EXTREAL1:th.24">EXTREAL1:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2"><b>suppose </b></a><a NAME="E1:46_1_2"/><i><font color="Green" title="E30">A3</font></i>: 
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 ;<br/><div class="add"><a NAME="E2:46_1_2"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="ref" href="extreal1.html#D3" title="EXTREAL1:def.3">EXTREAL1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:46_1_2"/>
(  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <font color="Maroon" title="c2">y</font> <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> <font color="Maroon" title="c2">y</font> )
 <b>by </b><i><a class="txt" href="extreal2.html#E1:46"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="extreal2.html#E1:46_1_2"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="extreal2.html#T47" target="_self" title="EXTREAL2:th.47">Th47</a>, <a class="ref" href="extreal1.html#T25" title="EXTREAL1:th.25">EXTREAL1:25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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