<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">q1</font>, <font color="Maroon" title="c5">q2</font>, <font color="Maroon" title="c6">r1</font>, <font color="Maroon" title="c7">r2</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"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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>  &amp; <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="c4">q1</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#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c6">r1</font> &amp;  <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>  <a href="int_3.html#R1" title="INT_3:pred.1">&lt;=</a> <font color="Maroon" title="c6">r1</font> &amp; <font color="Maroon" title="c6">r1</font> <a href="int_3.html#NR3" title="INT_3:NR.3">&lt;</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font> &amp; <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="c5">q2</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#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c7">r2</font> &amp;  <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>  <a href="int_3.html#R1" title="INT_3:pred.1">&lt;=</a> <font color="Maroon" title="c7">r2</font> &amp; <font color="Maroon" title="c7">r2</font> <a href="int_3.html#NR3" title="INT_3:NR.3">&lt;</a>  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c8">I</font> =  <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">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/>
<b>reconsider </b><font color="Maroon" title="c10">b'</font> = <font color="Maroon" title="c3">b</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/>
<b>reconsider </b><font color="Maroon" title="c11">q1'</font> = <font color="Maroon" title="c4">q1</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/>
<b>reconsider </b><font color="Maroon" title="c12">q2'</font> = <font color="Maroon" title="c5">q2</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/>
<b>reconsider </b><font color="Maroon" title="c13">r1'</font> = <font color="Maroon" title="c6">r1</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/>
<b>reconsider </b><font color="Maroon" title="c14">r2'</font> = <font color="Maroon" title="c7">r2</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/>
<a NAME="E8:23"/><i><font color="Green" title="E19">A2</font></i>: 
( <font color="Maroon" title="c9">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q1'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">b'</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">r1'</font> &amp; <font color="Maroon" title="c9">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">q2'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">b'</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">r2'</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c15">d</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="E2:23_1"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c15">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q1</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">b</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c16">d'</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E4:23_1"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c16">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q1'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">b'</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c17">e</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="E6:23_1"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">b</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c18">e'</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E8:23_1"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c18">e'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q2'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">b'</font>
 ;<br/>
<a NAME="E9:23_1"/><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c15">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">d'</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E2:23_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="int_3.html#E4:23_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="int_3.html#E6"><i><font color="Green" title="E4">Lm3</font></i></a></i>;<br/>
<a NAME="E10:23_1"/>
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">e'</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E6:23_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="int_3.html#E8:23_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="int_3.html#E6"><i><font color="Green" title="E4">Lm3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:23_1"/>
( <font color="Maroon" title="c9">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q1'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">b'</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">r1'</font> &amp; <font color="Maroon" title="c9">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">q2'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">b'</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c14">r2'</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E1:23"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="int_3.html#E2:23_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="int_3.html#E4:23_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="int_3.html#E6:23_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="int_3.html#E8:23_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="int_3.html#E9:23_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="int_3.html#E16"><i><font color="Green" title="E14">Lm9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:23"/><b>then </b><i><font color="Green" title="E20">A8</font></i>: 
( <font color="Maroon" title="c11">q1'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r1'</font> &amp; <font color="Maroon" title="c12">q2'</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r2'</font> )
 
;<br/>
<div><i><font color="Green" title="E21">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_2_1"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">r1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r2'</font> or <font color="Maroon" title="c13">r1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r2'</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_2_1_1"><b>case </b></a><a NAME="E1:23_2_1_1"/><i><font color="Green" title="E22">A10</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">r1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r2'</font>
 ;<br/><div class="add"><a NAME="E2:23_2_1_1"/><i><font color="Green" title="E23">A11</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">q2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">q1'</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r2'</font>
 <b>by </b><i><a class="txt" href="int_3.html#E9:23"><i><font color="Green" title="E20">A8</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_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_2_1_1_1_1"/>
( 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r2'</font> or 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">r1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r2'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:23_2_1_1_1_1_1"/>
0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">r1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r2'</font>
 ;<br/><div class="add"><a NAME="E2:23_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E24">A12</font></i>: 
( <font color="Maroon" title="c12">q2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">q1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="int_3.html#E2:23_2_1_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/><a NAME="E3:23_2_1_1_1_1_1"/>
<font color="Maroon" title="c12">q2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q1'</font>
 <b>by </b><i><a class="txt" href="int_3.html#E1:23"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="int_3.html#E2:23_2_1_1_1_1_1"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="int_3.html#E5"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:23_2_1_1_1_1_1"/>
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:23_2_1_1_1_1_2"/>
0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">r1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r2'</font>
 ;<br/><div class="add"><a NAME="E2:23_2_1_1_1_1_2"/><b>then </b>
0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">q2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">q1'</font>
 <b>by </b><i><a class="txt" href="int_3.html#E2:23_2_1_1"><i><font color="Green" title="E23">A11</font></i></a></i>;<br/><a NAME="E3:23_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E24">A13</font></i>: 
<a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">q2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">q1'</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="int_3.html#E15"><i><font color="Green" title="E13">Lm8</font></i></a></i>;<br/><a NAME="E4:23_2_1_1_1_1_2"/><i><font color="Green" title="E25">A14</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">q2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">q1'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><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="c10">b'</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <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="c10">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:23_2_1_1_1_1_2_1"/>
<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="c10">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="ref" href="int_3.html#D6" target="_self" title="INT_3:def.6">Def6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">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="c10">b'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">d'</font> = <font color="Maroon" title="c12">q2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">q1'</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">e</font> = <font color="Maroon" title="c5">q2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">q1</font></span>)</span> 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="E5:23_2_1_1_1_1_2_1"/><i><font color="Green" title="E26">A15</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c11">q1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c4">q1</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E9"><i><font color="Green" title="E7">Lm6</font></i></a></i>;<br/>
<font color="Maroon" title="c16">d'</font> = 
<font color="Maroon" title="c12">q2'</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c11">q1'</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="int_3.html#E5:23_2_1_1_1_1_2_1"><i><font color="Green" title="E26">A15</font></i></a>, <a class="txt" href="int_3.html#E16"><i><font color="Green" title="E14">Lm9</font></i></a></i>
;<br/>
<a NAME="E7:23_2_1_1_1_1_2_1"/><b>then </b>
<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="c16">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c17">e</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>then reconsider </b><font color="Maroon" title="c18">d</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="c16">d'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E9:23_2_1_1_1_1_2_1"/>
<font color="Maroon" title="c18">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c15">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c15">c</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E3:23_2_1_1_1_1_2"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="nat_1.html#T4" title="NAT_1:th.4">NAT_1:4</a></i>;<br/>
<b>hence </b><a NAME="E10:23_2_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">q2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">q1'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><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="c10">b'</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <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="c10">b'</font>
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c15">i1</font>, <font color="Maroon" title="c16">i2</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E6:23_2_1_1_1_1_2"/><i><font color="Green" title="E26">A16</font></i>: 
( <font color="Maroon" title="c15">i1</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="c16">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r2</font> &amp; <font color="Maroon" title="c15">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i2</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E1:23"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="int_3.html#D4" target="_self" title="INT_3:def.4">Def4</a></i>;<br/><a NAME="E7:23_2_1_1_1_1_2"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c14">r2'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 0
 <b>by </b><i><a class="txt" href="int_3.html#E6:23_2_1_1_1_1_2"><i><font color="Green" title="E26">A16</font></i></a>, <a class="txt" href="int_3.html#E5"><i><font color="Green" title="E3">Lm2</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><a NAME="E8:23_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">r1'</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c14">r2'</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">r1'</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E9:23_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E27">A17</font></i>: 
<font color="Maroon" title="c13">r1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r2'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">r1'</font>
 ;<br/><i><font color="Green" title="E28">A18</font></i>:  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font> = 
<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#T1" target="_self" title="INT_3:th.1">Th1</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="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>consider </b><font color="Maroon" title="c17">u</font>, <font color="Maroon" title="c18">v</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E12:23_2_1_1_1_1_2"/><i><font color="Green" title="E29">A19</font></i>: 
( <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r1</font> &amp; <font color="Maroon" title="c18">v</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="c3">b</font> &amp; <font color="Maroon" title="c17">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">v</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E1:23"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="int_3.html#D4" target="_self" title="INT_3:def.4">Def4</a></i>;<br/><a NAME="E13:23_2_1_1_1_1_2"/><i><font color="Green" title="E30">A20</font></i>: 
<font color="Maroon" title="c13">r1'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <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="c10">b'</font>
 <b>by </b><i><a class="txt" href="int_3.html#E1:23"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="int_3.html#E10:23_2_1_1_1_1_2"><i><font color="Green" title="E28">A18</font></i></a>, <a class="txt" href="int_3.html#E12:23_2_1_1_1_1_2"><i><font color="Green" title="E29">A19</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><font color="Maroon" title="c13">r1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r2'</font> = 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">q2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">q1'</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">b'</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E1:23_2_1_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="int_3.html#E2:23_2_1_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">q2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">q1'</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c10">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">q2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">q1'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c10">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/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">q2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">q1'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><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="c10">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/><b>hence </b><a NAME="E15:23_2_1_1_1_1_2"/>
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font>
 <b>by </b><i><a class="txt" href="int_3.html#E4:23_2_1_1_1_1_2"><i><font color="Green" title="E25">A14</font></i></a>, <a class="txt" href="int_3.html#E9:23_2_1_1_1_1_2"><i><font color="Green" title="E27">A17</font></i></a>, <a class="txt" href="int_3.html#E13:23_2_1_1_1_1_2"><i><font color="Green" title="E30">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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_2_1_1"/>
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_1_2"><b>case </b></a><a NAME="E1:23_2_1_2"/><i><font color="Green" title="E22">A21</font></i>: 
<font color="Maroon" title="c13">r1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r2'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:23_2_1_2"/><b>then </b><i><font color="Green" title="E23">A22</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r2'</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60" title="XREAL_1:th.60">XREAL_1:60</a></i>;<br/><a NAME="E3:23_2_1_2"/><i><font color="Green" title="E24">A23</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">r1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">r2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">r2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r1'</font>
 ;<br/><a NAME="E4:23_2_1_2"/><i><font color="Green" title="E25">A24</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">q1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">q2'</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">r2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r1'</font>
 <b>by </b><i><a class="txt" href="int_3.html#E8:23"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/><a NAME="E5:23_2_1_2"/><b>then </b>
0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">q1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">q2'</font>
 <b>by </b><i><a class="txt" href="int_3.html#E1:23_2_1_2"><i><font color="Green" title="E22">A21</font></i></a>, <a class="txt" href="int_3.html#E3:23_2_1_2"><i><font color="Green" title="E24">A23</font></i></a>, <a class="ref" href="xreal_1.html#T60" title="XREAL_1:th.60">XREAL_1:60</a></i>;<br/><a NAME="E6:23_2_1_2"/><b>then </b><i><font color="Green" title="E26">A25</font></i>: 
<a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">q1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">q2'</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="int_3.html#E15"><i><font color="Green" title="E13">Lm8</font></i></a></i>;<br/><a NAME="E7:23_2_1_2"/><i><font color="Green" title="E27">A26</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">q1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">q2'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><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="c10">b'</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <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="c10">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:23_2_1_2_1"/>
<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="c10">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="ref" href="int_3.html#D6" target="_self" title="INT_3:def.6">Def6</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">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="c10">b'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">d'</font> = <font color="Maroon" title="c11">q1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">q2'</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">e</font> = <font color="Maroon" title="c4">q1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">q2</font></span>)</span> 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="E5:23_2_1_2_1"/><i><font color="Green" title="E28">A27</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">q2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c5">q2</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E9"><i><font color="Green" title="E7">Lm6</font></i></a></i>;<br/>
<font color="Maroon" title="c16">d'</font> = 
<font color="Maroon" title="c11">q1'</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c12">q2'</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="int_3.html#E5:23_2_1_2_1"><i><font color="Green" title="E28">A27</font></i></a>, <a class="txt" href="int_3.html#E16"><i><font color="Green" title="E14">Lm9</font></i></a></i>
;<br/>
<a NAME="E7:23_2_1_2_1"/><b>then </b>
<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="c16">d'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c17">e</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>then reconsider </b><font color="Maroon" title="c18">d</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="c16">d'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E9:23_2_1_2_1"/>
<font color="Maroon" title="c18">d</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c15">c</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c15">c</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E6:23_2_1_2"><i><font color="Green" title="E26">A25</font></i></a>, <a class="ref" href="nat_1.html#T4" title="NAT_1:th.4">NAT_1:4</a></i>;<br/>
<b>hence </b><a NAME="E10:23_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">q1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">q2'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><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="c10">b'</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <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="c10">b'</font>
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c15">i1</font>, <font color="Maroon" title="c16">i2</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E9:23_2_1_2"/><i><font color="Green" title="E28">A28</font></i>: 
( <font color="Maroon" title="c15">i1</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="c16">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r1</font> &amp; <font color="Maroon" title="c15">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i2</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E1:23"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="int_3.html#D4" target="_self" title="INT_3:def.4">Def4</a></i>;<br/><a NAME="E10:23_2_1_2"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c13">r1'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> 0
 <b>by </b><i><a class="txt" href="int_3.html#E9:23_2_1_2"><i><font color="Green" title="E28">A28</font></i></a>, <a class="txt" href="int_3.html#E5"><i><font color="Green" title="E3">Lm2</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><a NAME="E11:23_2_1_2"/><b>then </b>
<font color="Maroon" title="c14">r2'</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c13">r1'</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r2'</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E12:23_2_1_2"/><b>then </b><i><font color="Green" title="E29">A29</font></i>: 
<font color="Maroon" title="c14">r2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r1'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">r2'</font>
 ;<br/><i><font color="Green" title="E30">A30</font></i>:  <a href="int_3.html#K2" title="INT_3:func.2">abs</a> <font color="Maroon" title="c3">b</font> = 
<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#T1" target="_self" title="INT_3:th.1">Th1</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="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>consider </b><font color="Maroon" title="c17">u</font>, <font color="Maroon" title="c18">v</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E15:23_2_1_2"/><i><font color="Green" title="E31">A31</font></i>: 
( <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r2</font> &amp; <font color="Maroon" title="c18">v</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="c3">b</font> &amp; <font color="Maroon" title="c17">u</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">v</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E1:23"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="int_3.html#D4" target="_self" title="INT_3:def.4">Def4</a></i>;<br/><a NAME="E16:23_2_1_2"/><i><font color="Green" title="E32">A32</font></i>: 
<font color="Maroon" title="c14">r2'</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <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="c10">b'</font>
 <b>by </b><i><a class="txt" href="int_3.html#E1:23"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="int_3.html#E13:23_2_1_2"><i><font color="Green" title="E30">A30</font></i></a>, <a class="txt" href="int_3.html#E15:23_2_1_2"><i><font color="Green" title="E31">A31</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><font color="Maroon" title="c14">r2'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">r1'</font> = 
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">q1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">q2'</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">b'</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E2:23_2_1_2"><i><font color="Green" title="E23">A22</font></i></a>, <a class="txt" href="int_3.html#E4:23_2_1_2"><i><font color="Green" title="E25">A24</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">q1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">q2'</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c10">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">q1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">q2'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c10">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/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K2" title="EUCLID:func.2">absreal</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">q1'</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">q2'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><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="c10">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/><b>hence </b><a NAME="E18:23_2_1_2"/>
<font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font>
 <b>by </b><i><a class="txt" href="int_3.html#E7:23_2_1_2"><i><font color="Green" title="E27">A26</font></i></a>, <a class="txt" href="int_3.html#E12:23_2_1_2"><i><font color="Green" title="E29">A29</font></i></a>, <a class="txt" href="int_3.html#E16:23_2_1_2"><i><font color="Green" title="E32">A32</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<font color="Maroon" title="c6">r1</font> = 
<font color="Maroon" title="c6">r1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><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>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
<font color="Maroon" title="c6">r1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q1</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#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q1</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q2</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#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c7">r2</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q2</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E1:23"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="int_3.html#E10:23"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c7">r2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q2</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#K4" title="RLVECT_1:func.4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">q2</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c7">r2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <span class="p1">(<span class="default"><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>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon" title="c7">r2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E12:23"/>
( <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font> &amp; <font color="Maroon" title="c6">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r2</font> )
 <b>by </b><i><a class="txt" href="int_3.html#E10:23"><i><font color="Green" title="E21">A9</font></i></a></i>;<br/>


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