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

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">q1</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">q2</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">r1</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> , <font color="Maroon">r2</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> ;<br/>

<b>assume </b><a NAME="E1:23"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r1</font> &amp;  <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a>  <a href="int_3.html#R1" target="_self">&lt;=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="int_3.html#NR3" target="_self">&lt;</a>  <a href="int_3.html#K2" target="_self">abs</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r2</font> &amp;  <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a>  <a href="int_3.html#R1" target="_self">&lt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r2</font> <a href="int_3.html#NR3" target="_self">&lt;</a>  <a href="int_3.html#K2" target="_self">abs</a> <font color="Maroon">b</font> )
 ;<br/>

<b>set </b><font color="Maroon">I</font> =  <a href="int_3.html#K1" target="_self">INT.Ring</a> ;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q1'</font> = <font color="Maroon">q1</font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q2'</font> = <font color="Maroon">q2</font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r1'</font> = <font color="Maroon">r1</font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r2'</font> = <font color="Maroon">r2</font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<a NAME="E8:23"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q1'</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b'</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r1'</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q2'</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b'</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">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">d</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> <b> such that </b><br/><a NAME="E2:23_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font>
 ;<br/>
<b>consider </b><font color="Maroon">d'</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:23_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">d'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1'</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b'</font>
 ;<br/>
<b>consider </b><font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a> <b> such that </b><br/><a NAME="E6:23_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font>
 ;<br/>
<b>consider </b><font color="Maroon">e'</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E8:23_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">e'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2'</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b'</font>
 ;<br/>
<a NAME="E9:23_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d'</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E10:23_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e'</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E8:23"><i><font color="Green">E22</font></i></a>, <a class="txt" href="int_3.html#E2:23_1"><i><font color="Green">E23</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E11:23_1"/>
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q1'</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b'</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r1'</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q2'</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b'</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r2'</font> )
 <b>by </b><i>, , , <a class="txt" href="int_3.html#E8:23"><i><font color="Green">E22</font></i></a>, <a class="txt" href="int_3.html#E2:23_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="int_3.html#E4:23_1"><i><font color="Green">E24</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:23"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">q1'</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1'</font> &amp; <font color="Maroon">q2'</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2'</font> )
 
;<br/>
<div><i><font color="Green">E46</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">&lt;=</a> <font color="Maroon">r1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2'</font> or <font color="Maroon">r1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2'</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="23_2_1_1"><b>case </b></a><a NAME="E1:23_2_1_1"/><i><font color="Green">E47</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2'</font>
 ;<br/><div class="add"><a NAME="E2:23_2_1_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q2'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q1'</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2'</font>
 <b>by </b><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">=</a> <font color="Maroon">r1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2'</font> or 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">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">=</a> <font color="Maroon">r1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2'</font>
 ;<br/><div class="add"><a NAME="E2:23_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">q2'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q1'</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><a NAME="E3:23_2_1_1_1_1_1"/>
<font color="Maroon">q2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1'</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:23_2_1_1_1_1_1"/>
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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" target="_self">&lt;&gt;</a> <font color="Maroon">r1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">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" target="_self">&lt;&gt;</a> <font color="Maroon">q2'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q1'</font>
 <b>by </b><i/>;<br/><a NAME="E3:23_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">q2'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q1'</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i/>;<br/><a NAME="E4:23_2_1_1_1_1_2"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">q2'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q1'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">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">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">c</font> = <a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">d'</font> = <font color="Maroon">q2'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q1'</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">q2</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">q1</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a>  ;<br/>
<a NAME="E5:23_2_1_1_1_1_2_1"/><i><font color="Green">E54</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">q1'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">q1</font>
 
<b>by </b><i/>;<br/>
<font color="Maroon">d'</font> = 
<font color="Maroon">q2'</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">q1'</font></span>)</span>

<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="txt" href="int_3.html#E4"><i><font color="Green">Lemma26</font></i></a>, </i>
;<br/>
<a NAME="E7:23_2_1_1_1_1_2_1"/><b>then </b>
<a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">d'</font> <a href="hidden.html#R1">=</a> <a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">d</font> = <a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">d'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E9:23_2_1_1_1_1_2_1"/>
<font color="Maroon">d</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T20">NAT_1:20</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">absreal</a>  <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">q2'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q1'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font>
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">i1</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">i2</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E6:23_2_1_1_1_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a>  &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E7:23_2_1_1_1_1_2"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r2'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binop_2.html#K7">-</a> 0
 <b>by </b><i><a class="txt" href="int_3.html#E4"><i><font color="Green">Lemma26</font></i></a>, , <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E8:23_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon">r1'</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r2'</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1'</font> <a href="xcmplx_0.html#K2">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:23_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">r1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1'</font>
 ;<br/><i><font color="Green">E57</font></i>:  <a href="int_3.html#K2" target="_self">abs</a> <font color="Maroon">b</font> = 
<a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i/>
;<br/><b>consider </b><font color="Maroon">u</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">v</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E12:23_2_1_1_1_1_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="int_3.html#K2" target="_self">abs</a> <font color="Maroon">b</font> &amp; <font color="Maroon">u</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E13:23_2_1_1_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">r1'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font>
 <b>by </b><i>, , <a class="txt" href="int_3.html#E5"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><font color="Maroon">r1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2'</font> = 
 <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q2'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q1'</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i><a class="txt" href="int_3.html#E8:23"><i><font color="Green">E22</font></i></a>, , <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p2">(<span class="default"><font color="Maroon">q2'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q1'</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">q2'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q1'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#D2">EUCLID:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">q2'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q1'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#D2">EUCLID:def 2</a></i>
;<br/><b>hence </b><a NAME="E15:23_2_1_1_1_1_2"/>
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 <b>by </b><i>, , <a class="txt" href="int_3.html#E6"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="xxreal_0.html#T2">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">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">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">E60</font></i>: 
<font color="Maroon">r1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:23_2_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">r1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2'</font></span>)</span> <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/><a NAME="E3:23_2_1_2"/><i><font color="Green">E101</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">r1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r2'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r2'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1'</font>
 ;<br/><a NAME="E4:23_2_1_2"/><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">q1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q2'</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1'</font>
 <b>by </b><i><a class="txt" href="int_3.html#E1:23"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E5:23_2_1_2"/><b>then </b>
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q2'</font>
 <b>by </b><i><a class="txt" href="int_3.html#E8:23"><i><font color="Green">E22</font></i></a>, , <a class="ref" href="xreal_1.html#T60">XREAL_1:60</a></i>;<br/><a NAME="E6:23_2_1_2"/><b>then </b><i><font color="Green">E103</font></i>: 
<a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">q1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q2'</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i/>;<br/><a NAME="E7:23_2_1_2"/><i><font color="Green">E104</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">q1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q2'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">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">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">c</font> = <a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">d'</font> = <font color="Maroon">q1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q2'</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">q1</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">q2</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <a href="int_3.html#K1" target="_self">INT.Ring</a>  ;<br/>
<a NAME="E5:23_2_1_2_1"/><i><font color="Green">E105</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">q2'</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <font color="Maroon">q2</font>
 
<b>by </b><i/>;<br/>
<font color="Maroon">d'</font> = 
<font color="Maroon">q1'</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">q2'</font></span>)</span>

<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i>, </i>
;<br/>
<a NAME="E7:23_2_1_2_1"/><b>then </b>
<a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">d'</font> <a href="hidden.html#R1">=</a> <a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">d</font> = <a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">d'</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E9:23_2_1_2_1"/>
<font color="Maroon">d</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K2">*</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="txt" href="int_3.html#E4"><i><font color="Green">Lemma26</font></i></a>, <a class="ref" href="nat_1.html#T20">NAT_1:20</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">absreal</a>  <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">q1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q2'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font>
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">i1</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">i2</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E9:23_2_1_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a>  &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E10:23_2_1_2"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r1'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="binop_2.html#K7">-</a> 0
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E11:23_2_1_2"/><b>then </b>
<font color="Maroon">r2'</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r1'</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2'</font> <a href="xcmplx_0.html#K2">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E12:23_2_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">r2'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2'</font>
 ;<br/><i><font color="Green">E108</font></i>:  <a href="int_3.html#K2" target="_self">abs</a> <font color="Maroon">b</font> = 
<a href="int_3.html#K3" target="_self">absint</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i/>
;<br/><b>consider </b><font color="Maroon">u</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">v</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E15:23_2_1_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="int_3.html#K2" target="_self">abs</a> <font color="Maroon">b</font> &amp; <font color="Maroon">u</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E16:23_2_1_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">r2'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font>
 <b>by </b><i>, <a class="txt" href="int_3.html#E6"><i><font color="Green">Lemma30</font></i></a>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><font color="Maroon">r2'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">r1'</font> = 
 <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q2'</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p2">(<span class="default"><font color="Maroon">q1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q2'</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">q1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q2'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#D2">EUCLID:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">q1'</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">q2'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="euclid.html#K2">absreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#D2">EUCLID:def 2</a></i>
;<br/><b>hence </b><a NAME="E18:23_2_1_2"/>
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 <b>by </b><i>, <a class="txt" href="int_3.html#E5"><i><font color="Green">Lemma29</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<font color="Maroon">r1</font> = 
<font color="Maroon">r1</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a> </span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<font color="Maroon">r1</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">q1</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">r2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">q2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">r2</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">q2</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">r2</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="int_3.html#K1" target="_self">INT.Ring</a> </span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon">r2</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<b>hence </b><a NAME="E12:23"/>
( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> )
 <b>by </b><i/>;<br/>


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