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<b>let </b><font color="Maroon" title="c2">a</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>reconsider </b><font color="Maroon" title="c3">a'</font> = <font color="Maroon" title="c2">a</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<i><font color="Green" title="E13">A1</font></i>: <a href="int_3.html#K3" title="INT_3:func.3">absint</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font> = 
<a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a'</font>
<b>by </b><i><a class="ref" href="int_3.html#D6" target="_self" title="INT_3:def.6">Def6</a></i>
<br/>.= 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">a'</font>
<b>by </b><i><a class="ref" href="euclid.html#D2" title="EUCLID:def.2">EUCLID:def 2</a></i>
;<br/>
<a NAME="E3:18"/><i><font color="Green" title="E14">A2</font></i>: 
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_1"/>
( <font color="Maroon" title="c2">a</font> <a href="int_3.html#NR2" title="INT_3:NR.2">&gt;=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  or  not <font color="Maroon" title="c2">a</font> <a href="int_3.html#NR2" title="INT_3:NR.2">&gt;=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_1"><b>case </b></a><a NAME="E1:18_2_1_1"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="int_3.html#NR2" title="INT_3:NR.2">&gt;=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a> 
 ;<br/><div class="add"><a NAME="E2:18_2_1_1"/><b>then </b><i><font color="Green" title="E16">A4</font></i>: 
 <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="int_3.html#D5" target="_self" title="INT_3:def.5">Def5</a></i>;<br/><b>consider </b><font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E4:18_2_1_1"/><i><font color="Green" title="E17">A5</font></i>: 
( <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  &amp; <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">d</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E1:18_2_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="int_3.html#D4" target="_self" title="INT_3:def.4">Def4</a></i>;<br/><a NAME="E5:18_2_1_1"/>
0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="int_3.html#E3:18"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a>, <a class="txt" href="int_3.html#E4:18_2_1_1"><i><font color="Green" title="E17">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E6:18_2_1_1"/>
<a href="int_3.html#K3" title="INT_3:func.3">absint</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="int_3.html#E2:18"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="int_3.html#E2:18_2_1_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="int_3.html#E4:18_2_1_1"><i><font color="Green" title="E17">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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_1_2"><b>case </b></a><a NAME="E1:18_2_1_2"/><i><font color="Green" title="E15">A6</font></i>: 
not <font color="Maroon" title="c2">a</font> <a href="int_3.html#NR2" title="INT_3:NR.2">&gt;=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a> 
 ;<br/><div class="add"><a NAME="E2:18_2_1_2"/>
 ex <font color="Olive" title="b1">c'</font>, <font color="Olive" title="b2">d'</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  &amp; <font color="Olive" title="b2">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1_2_1"><b>proof </b></a><div class="add">

<b>take </b>
0
;<br/>

<b>take </b>
<font color="Maroon" title="c3">a'</font>
;<br/>

<b>thus </b><a NAME="E1:18_2_1_2_1"/>
( 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  &amp; <font color="Maroon" title="c3">a'</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#E5"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c4">c'</font>, <font color="Maroon" title="c5">d'</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E4:18_2_1_2"/><i><font color="Green" title="E16">A7</font></i>: 
( <font color="Maroon" title="c4">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a>  &amp; <font color="Maroon" title="c5">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 ;<br/><a NAME="E5:18_2_1_2"/><i><font color="Green" title="E17">A8</font></i>: 
not 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">d'</font>
 <b>by </b><i><a class="txt" href="int_3.html#E1:18_2_1_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="int_3.html#E4:18_2_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="int_3.html#D4" target="_self" title="INT_3:def.4">Def4</a>, <a class="txt" href="int_3.html#E5"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/><a href="int_3.html#K3" title="INT_3:func.3">absint</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font> = 
<a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d'</font>
<b>by </b><i><a class="txt" href="int_3.html#E4:18_2_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="int_3.html#D6" target="_self" title="INT_3:def.6">Def6</a></i>
<br/>.= 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c5">d'</font>
<b>by </b><i><a class="ref" href="euclid.html#D2" title="EUCLID:def.2">EUCLID:def 2</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c5">d'</font>
<b>by </b><i><a class="txt" href="int_3.html#E5:18_2_1_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="int_3.html#E4:18_2_1_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="int_3.html#E9"><i><font color="Green" title="E7">Lm6</font></i></a></i>
<br/>.= 
 <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="int_3.html#E1:18_2_1_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="int_3.html#D5" target="_self" title="INT_3:def.5">Def5</a></i>
;<br/><b>hence </b><a NAME="E7:18_2_1_2"/>
<a href="int_3.html#K3" title="INT_3:func.3">absint</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c2">a</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:18"/>
<a href="int_3.html#K3" title="INT_3:func.3">absint</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c2">a</font>
 ;<br/>


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