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<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> or  not 0 <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="39_1_1"><b>suppose </b></a><a NAME="E1:39_1_1"/><i><font color="Green" title="E26">A2</font></i>: 
0 <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="E2:39_1_1"/><b>then </b><i><font color="Green" title="E27">A3</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="extreal2.html#E1:39"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="extreal1.html#D3" title="EXTREAL1:def.3">EXTREAL1:def 3</a></i>;<br/><a NAME="E3:39_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="extreal2.html#E1:39"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="extreal2.html#E1:39_1_1"><i><font color="Green" title="E26">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:39_1_1"/>
<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="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="extreal2.html#E2:39_1_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2"><b>suppose </b></a><a NAME="E1:39_1_2"/><i><font color="Green" title="E26">A4</font></i>: 
not 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E27">A5</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="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#T37" title="EXTREAL1:th.37">EXTREAL1:37</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="extreal2.html#E1:39"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="supinf_2.html#T3" title="SUPINF_2:th.3">SUPINF_2:3</a></i>
;<br/><a NAME="E3:39_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal2.html#E1:39"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="extreal2.html#E1:39_1_2"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E4:39_1_2"/>
<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="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="extreal2.html#E2:39_1_2"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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