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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_1_1"/>
 <a href="complex1.html#NK18" target="_self">abs</a> <font color="Maroon">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_2"/><b>then </b><i><font color="Green">E12</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:1_1_1_1_2"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_1_2"/>
 <a href="complex1.html#NK18" target="_self">abs</a> <font color="Maroon">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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