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

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">xx</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>





<b>assume </b><a NAME="E1:22"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E2:22"/><b>then </b><i><font color="Green">E30</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
;<br/>
<b>set </b><font color="Maroon">i</font> = <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>;<br/>
<a NAME="E3:22"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E4:22"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="int_2.html#D3">INT_2:def 3</a></i>;<br/>
<a NAME="E5:22"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E6:22"/><b>then </b><i><font color="Green">E45</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/>
<a NAME="E7:22"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="newton.html#T71">NEWTON:71</a></i>;<br/>
<a NAME="E8:22"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a>  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1"/>
( <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_1_1"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:22_1_1_1"/>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a></i>;<br/><a NAME="E4:22_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/><b>hence </b><a NAME="E5:22_1_1_1"/>
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a>  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="euler_1.html#E3"><i><font color="Green">Lemma29</font></i></a>, <a class="txt" href="euler_1.html#E3:22"><i><font color="Green">E31</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_1_2"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:22_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <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/><a NAME="E3:22_1_1_2"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a></i>;<br/><a NAME="E4:22_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/><b>hence </b><a NAME="E5:22_1_1_2"/>
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a>  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="euler_1.html#E3"><i><font color="Green">Lemma29</font></i></a>, <a class="txt" href="euler_1.html#E3:22"><i><font color="Green">E31</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E9:22"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="nat_1.html#R1">divides</a>  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> &amp; <font color="Olive">m</font> <a href="nat_1.html#R1">divides</a>  <a href="int_2.html#K1">abs</a> <font color="Maroon">b</font> holds <br/><font color="Olive">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">mm</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:22_2"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">mm</font> <a href="nat_1.html#R1">divides</a>  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> &amp; <font color="Maroon">mm</font> <a href="nat_1.html#R1">divides</a>  <a href="int_2.html#K1">abs</a> <font color="Maroon">b</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:22_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mm</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">mm</font> = <font color="Maroon">mm</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_2_1_1"/>
( <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <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="22_2_1_1_1"><b>suppose </b></a><a NAME="E1:22_2_1_1_1"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:22_2_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_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="E1:22_2_1_1_1_1_1"/>
( <font color="Maroon">mm</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">mm</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_2_1_1_1_1_1_1"/>
<font color="Maroon">mm</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:22_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">mm</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="euler_1.html#E3:22"><i><font color="Green">E31</font></i></a>, , </i>;<br/><b>then consider </b><font color="Maroon">t</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:22_2_1_1_1_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">mm</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/><i><font color="Green">E76</font></i>: <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon">mm</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">mm</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="euler_1.html#E4:22"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xcmplx_1.html#T26">XCMPLX_1:26</a></i>
<br/>.= 
<font color="Maroon">mm</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>

;<br/><a NAME="E6:22_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T136">XREAL_1:136</a></i>;<br/><b>then reconsider </b><font color="Maroon">tt</font> = <font color="Maroon">t</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E8:22_2_1_1_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mm</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">tt</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E4"><i><font color="Green">Lemma35</font></i></a></i>;<br/><b>hence </b><a NAME="E9:22_2_1_1_1_1_1_1"/>
<font color="Maroon">mm</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_2_1_1_1_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">mm</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:22_2_1_1_1_1_1_2"/><b>then </b>
 <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="euler_1.html#E3:22"><i><font color="Green">E31</font></i></a>, <a class="ref" href="int_2.html#T3">INT_2:3</a></i>;<br/><b>hence </b><a NAME="E3:22_2_1_1_1_1_1_2"/>
<font color="Maroon">mm</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="txt" href="euler_1.html#E4:22"><i><font color="Green">E34</font></i></a>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:22_2_1_1_1"/>
<font color="Maroon">mm</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_2"><b>suppose </b></a><a NAME="E1:22_2_1_1_2"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:22_2_1_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_2_1_1_2_1_1"/>
( <font color="Maroon">mm</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">mm</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:22_2_1_1_2_1_1_1"/>
<font color="Maroon">mm</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:22_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">mm</font> <a href="int_1.html#R2">divides</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="euler_1.html#E3:22"><i><font color="Green">E31</font></i></a>, , </i>;<br/><b>then consider </b><font color="Maroon">t</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:22_2_1_1_2_1_1_1"/><i><font color="Green">E91</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">mm</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/><i><font color="Green">E92</font></i>: <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><font color="Maroon">mm</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">mm</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="euler_1.html#E4:22"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xcmplx_1.html#T26">XCMPLX_1:26</a></i>
<br/>.= 
<font color="Maroon">mm</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>

;<br/><a NAME="E6:22_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T136">XREAL_1:136</a></i>;<br/><b>then reconsider </b><font color="Maroon">tt</font> = <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E8:22_2_1_1_2_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">mm</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">tt</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E4"><i><font color="Green">Lemma35</font></i></a></i>;<br/><b>hence </b><a NAME="E9:22_2_1_1_2_1_1_1"/>
<font color="Maroon">mm</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:22_2_1_1_2_1_1_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">mm</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:22_2_1_1_2_1_1_2"/><b>then </b>
 <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="euler_1.html#E3:22"><i><font color="Green">E31</font></i></a>, <a class="ref" href="int_2.html#T3">INT_2:3</a></i>;<br/><b>hence </b><a NAME="E3:22_2_1_1_2_1_1_2"/>
<font color="Maroon">mm</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="txt" href="euler_1.html#E4:22"><i><font color="Green">E34</font></i></a>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:22_2_1_1_2"/>
<font color="Maroon">mm</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:22_2"/>
<font color="Maroon">mm</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="txt" href="euler_1.html#E3:22"><i><font color="Green">E31</font></i></a>, <a class="ref" href="newton.html#T63">NEWTON:63</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:22"/><b>then </b>
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="txt" href="euler_1.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="txt" href="euler_1.html#E2:22"><i><font color="Green">E30</font></i></a>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:22"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">xx</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="int_2.html#K3">gcd</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="int_2.html#D3">INT_2:def 3</a></i>;<br/>


</div>
