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

<b>let </b><font color="Maroon" title="c1">m1</font>, <font color="Maroon" title="c2">m2</font>, <font color="Maroon" title="c3">c2</font>, <font color="Maroon" title="c4">c1</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>







<b>assume </b><a NAME="E1:58"/><i><font color="Green" title="E42">A1</font></i>: 
( <font color="Maroon" title="c1">m1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">m2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">c2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font> )
 ;<br/>

<a NAME="E2:58"/><i><font color="Green" title="E43">A2</font></i>: 
( <font color="Maroon" title="c1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> &amp; <font color="Maroon" title="c2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="int_4.html#E51"><i><font color="Green" title="E36">Lm11</font></i></a>, <a class="ref" href="int_2.html#T31" title="INT_2:th.31">INT_2:31</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">d</font> = <font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font>;<br/>
<b>set </b><font color="Maroon" title="c6">d1</font> = <font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">d2</font> = <font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">d3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">c2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span>;<br/>
<a NAME="E3:58"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="int_4.html#E1:58"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="int_2.html#T35" title="INT_2:th.35">INT_2:35</a></i>;<br/>
<a NAME="E4:58"/><i><font color="Green" title="E45">A4</font></i>: 
<font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span>,<font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="int_4.html#E1:58"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="int_4.html#E52"><i><font color="Green" title="E37">Lm12</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">r</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E6:58"/><i><font color="Green" title="E46">A5</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">c2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="int_4.html#T15" target="_self" title="INT_4:th.15">Th15</a></i>;<br/>
<a NAME="E7:58"/><i><font color="Green" title="E47">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">c2</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">c1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">m1</font> <a href="int_2.html#NK3" title="INT_2:NK.3">lcm'</a> <font color="Maroon" title="c2">m2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:58_1"/><i><font color="Green" title="E48">A7</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">c2</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>

<i><font color="Green" title="E49">A8</font></i>: <font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">m1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">m1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">m1</font></span>)</span>
<b>by </b><i><a class="txt" href="int_4.html#E1:58"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="int_1.html#T86" title="INT_1:th.86">INT_1:86</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">m1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">m1</font>
<b>by </b><i><a class="txt" href="int_4.html#E1:58_1"><i><font color="Green" title="E48">A7</font></i></a></i>
;<br/>
<a NAME="E3:58_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">m1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font></span>)</span>
 
;<br/>
<a NAME="E4:58_1"/><b>then </b><i><font color="Green" title="E50">A9</font></i>: 
<font color="Maroon" title="c1">m1</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">c1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">m1</font>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">c2</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m2</font>
<b>by </b><i><a class="txt" href="int_4.html#E2:58_1"><i><font color="Green" title="E49">A8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">c2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="int_4.html#E1:58"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="int_4.html#E2:58"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="int_4.html#E51"><i><font color="Green" title="E36">Lm11</font></i></a></i>
;<br/>
<a NAME="E6:58_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">c2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E1:58"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="int_4.html#E2:58"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="int_4.html#E1:58_1"><i><font color="Green" title="E48">A7</font></i></a>, <a class="txt" href="int_4.html#E50"><i><font color="Green" title="E35">Lm10</font></i></a></i>;<br/>
<a NAME="E7:58_1"/><b>then </b>
<font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E3:58"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="int_4.html#T7" target="_self" title="INT_4:th.7">Th7</a></i>;<br/>
<a NAME="E8:58_1"/><b>then </b><i><font color="Green" title="E51">A10</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">m1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">c2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="int_4.html#E50"><i><font color="Green" title="E35">Lm10</font></i></a></i>;<br/>
<a NAME="E9:58_1"/>
<font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="int_4.html#E1:58"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="int_4.html#E2:58"><i><font color="Green" title="E43">A2</font></i></a></i>;<br/>
<a NAME="E10:58_1"/><b>then </b>
<font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">m1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E4:58"><i><font color="Green" title="E45">A4</font></i></a>, <a class="txt" href="int_4.html#E6:58"><i><font color="Green" title="E46">A5</font></i></a>, <a class="txt" href="int_4.html#E8:58_1"><i><font color="Green" title="E51">A10</font></i></a>, <a class="ref" href="int_4.html#T26" target="_self" title="INT_4:th.26">Th26</a></i>;<br/>
<a NAME="E11:58_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">m1</font></span>)</span>,<font color="Maroon" title="c9">r</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="int_4.html#K1" title="INT_4:func.1">Cong</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T26" title="EQREL_1:th.26">EQREL_1:26</a></i>;<br/>
<a NAME="E12:58_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">m1</font>,<font color="Maroon" title="c9">r</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_4.html#D1" target="_self" title="INT_4:def.1">Def1</a></i>;<br/>
<a NAME="E13:58_1"/><b>then </b>
<font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">m1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">w</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E15:58_1"/><i><font color="Green" title="E52">A11</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">m1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">w</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<font color="Maroon" title="c10">x</font> = 
<font color="Maroon" title="c4">c1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c1">m1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">m1</font></span>)</span>
<b>by </b><i><a class="txt" href="int_4.html#E2:58_1"><i><font color="Green" title="E49">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">c1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">w</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">m1</font></span>)</span>
<b>by </b><i><a class="txt" href="int_4.html#E15:58_1"><i><font color="Green" title="E52">A11</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">w</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">m2</font></span>)</span>
<b>by </b><i><a class="txt" href="int_4.html#E2:58"><i><font color="Green" title="E43">A2</font></i></a></i>
;<br/>
<a NAME="E17:58_1"/><b>then </b>
<font color="Maroon" title="c2">m2</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">c1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E18:58_1"/><b>then </b>
<font color="Maroon" title="c1">m1</font> <a href="int_2.html#NK3" title="INT_2:NK.3">lcm'</a> <font color="Maroon" title="c2">m2</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c10">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">c1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_4.html#E4:58_1"><i><font color="Green" title="E50">A9</font></i></a>, <a class="ref" href="int_2.html#T27" title="INT_2:th.27">INT_2:27</a></i>;<br/>
<b>hence </b><a NAME="E19:58_1"/>
<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c4">c1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">m1</font> <a href="int_2.html#NK3" title="INT_2:NK.3">lcm'</a> <font color="Maroon" title="c2">m2</font>
 <b>by </b><i><a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c9">r</font>
;<br/>

<b>thus </b><a NAME="E8:58"/>
( ( for <font color="Olive" title="b1">x</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">c2</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">c1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">m1</font> <a href="int_2.html#NK3" title="INT_2:NK.3">lcm'</a> <font color="Maroon" title="c2">m2</font> ) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">r</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">c2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c2">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="int_4.html#E6:58"><i><font color="Green" title="E46">A5</font></i></a>, <a class="txt" href="int_4.html#E7:58"><i><font color="Green" title="E47">A6</font></i></a></i>;<br/>


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