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<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a> ;<br/>

<b>thus </b><a NAME="E1:12_1_2"/>
( <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  implies ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c2">a</font> iff <font color="Maroon" title="c3">x</font> <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="int_3.html#E6"><i><font color="Green" title="E4">Lm3</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c4">b</font> = <font color="Maroon" title="c2">a</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>assume </b><a NAME="E3:12_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a> 
 ;<br/>

<b>hence </b><a NAME="E4:12_1_2"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#NK19" title="COMPLEX1:NK.19">abs</a> <font color="Maroon" title="c2">a</font> iff <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E6"><i><font color="Green" title="E4">Lm3</font></i></a>, <a class="ref" href="complex1.html#T156" title="COMPLEX1:th.156">COMPLEX1:156</a></i>;<br/>


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