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<b>take </b>
 <a href="int_3.html#K3" title="INT_3:func.3">absint</a> 
; <i><font color="Red">:: according to </font></i><a class="ref" href="int_3.html#D9" target="_self" title="INT_3:def.9">INT_3:def 9</a><br/>

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</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>assume </b><a NAME="E1:23_1_1"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">b'</font> = <font color="Maroon" title="c3">b</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_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="E1:23_1_1_1_1"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b'</font> or <font color="Maroon" title="c4">b'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1_1"><b>case </b></a><a NAME="E1:23_1_1_1_1_1"/>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">b'</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">r</font> being  <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>  st <br/>( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">r</font> &amp; ( <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>  or <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="Olive" title="b2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <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="c3">b</font> ) )
 <b>by </b><i><a class="txt" href="int_3.html#E1:23_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="int_3.html#E15"><i><font color="Green" title="E10">Lm7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1_1_2"><b>case </b></a><a NAME="E1:23_1_1_1_1_2"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c4">b'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:23_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E18">A3</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c4">b'</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T60" title="XREAL_1:th.60">XREAL_1:60</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">c</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c4">b'</font> as   <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>  <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">q</font>, <font color="Maroon" title="c7">r</font> being   <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> <b> such that </b><br/><a NAME="E5:23_1_1_1_1_2"/><i><font color="Green" title="E19">A4</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">q</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">r</font> &amp; ( <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>  or <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="c7">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <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="c5">c</font> ) )
 <b>by </b><i><a class="txt" href="int_3.html#E2:23_1_1_1_1_2"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="int_3.html#E6"><i><font color="Green" title="E4">Lm3</font></i></a>, <a class="txt" href="int_3.html#E15"><i><font color="Green" title="E10">Lm7</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">t</font> =  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">q</font> as   <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/><a NAME="E7:23_1_1_1_1_2"/><i><font color="Green" title="E20">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">t</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">r</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#E5:23_1_1_1_1_2"><i><font color="Green" title="E19">A4</font></i></a></i>;<br/><a NAME="E8:23_1_1_1_1_2"/>
( <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>  or <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="c7">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <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="c3">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_1_1_1_2_1"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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> 
 ;<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="c5">c</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">c</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> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c4">b'</font></span>)</span>
<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="c4">b'</font>
<b>by </b><i><a class="txt" href="int_3.html#E2:23_1_1_1_1_2"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c4">b'</font>
<b>by </b><i><a class="txt" href="int_3.html#E1:23_1_1_1_1_2"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<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="c4">b'</font>
<b>by </b><i><a class="ref" href="euclid.html#D2" title="EUCLID:def.2">EUCLID:def 2</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="c3">b</font>
<b>by </b><i><a class="ref" href="int_3.html#D6" target="_self" title="INT_3:def.6">Def6</a></i>
;<br/>
<b>hence </b><a NAME="E3:23_1_1_1_1_2_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="c7">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <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="c3">b</font>
 <b>by </b><i><a class="txt" href="int_3.html#E5:23_1_1_1_1_2"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="int_3.html#E1:23_1_1_1_1_2_1"><i><font color="Green" title="E21">A6</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:23_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">r</font> being  <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>  st <br/>( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">r</font> &amp; ( <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>  or <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="Olive" title="b2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <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="c3">b</font> ) )
 <b>by </b><i><a class="txt" href="int_3.html#E7:23_1_1_1_1_2"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:23_1_1"/>
 ex <font color="Olive" title="b1">q</font>, <font color="Olive" title="b2">r</font> being  <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>  st <br/>( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">r</font> &amp; ( <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>  or <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="Olive" title="b2">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <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="c3">b</font> ) )
 ;<br/>


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