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

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>







<b>assume </b><a NAME="E1:9"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Maroon">a</font>,<font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> )
 ;<br/>

<b>assume </b><a NAME="E2:9"/>
not <font color="Maroon">m</font>,<font color="Maroon">b</font> <a href="int_2.html#R2">are_relative_prime</a> 
 ;<br/>

<a NAME="E3:9"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>set </b><font color="Maroon">k</font> = <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font>;<br/>
<a NAME="E4:9"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E5:9"/>
( <font color="Maroon">m</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 )
 
<b>by </b><i/>;<br/>
<a NAME="E6:9"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">m</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="E7:9"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">m</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">m</font> &amp; <font color="Maroon">m</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/>
<b>then consider </b><font color="Maroon">km</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:9"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">km</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E10:9"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">km</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">km</font> = <font color="Maroon">km</font> as    <a href="subset_1.html#M2">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/>
<b>consider </b><font color="Maroon">kb</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E13:9"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kb</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E15:9"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kb</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">km</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kb</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">km</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kb</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">km</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">km</font> )
 <b>by </b><i>, , <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<a NAME="E16:9"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kb</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">km</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</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="xcmplx_0.html#K3">*</a> <font color="Maroon">kb</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">km</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">tm</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kb</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">km</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span>;<br/>
<a NAME="E17:9"/><i><font color="Green">E59</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kb</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">km</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kb</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">km</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:9_1"/><i><font color="Green">E61</font></i>: 
0 <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> 0
 
<b>by </b><i><a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/>
<a NAME="E3:9_1"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">km</font> <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">km</font></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">km</font></span>)</span> <a href="int_2.html#K3">gcd</a> 0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">km</font></span>)</span> <a href="nat_1.html#K10">hcf</a> 0
<b>by </b><i>, <a class="ref" href="int_2.html#D3">INT_2:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">km</font>
<b>by </b><i><a class="ref" href="newton.html#T65">NEWTON:65</a></i>
;<br/>
<b>hence </b><a NAME="E5:9_1"/>
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i>, , , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:9"/><i><font color="Green">E62</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kb</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">km</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 implies <font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_2"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kb</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">km</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>assume </b><a NAME="E2:9_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E3:9_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xcmplx_0.html#K4">-</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E4:9_2"/>
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</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"><font color="Maroon">km</font> <a href="int_2.html#K3">gcd</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">kb</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">km</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , , , , , , <a class="ref" href="euler_1.html#T16">EULER_1:16</a></i>;<br/>
<b>hence </b><a NAME="E5:9_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="int_1.html#T22">INT_1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:9"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <font color="Maroon">m</font></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">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_2.html#D3">INT_2:def 3</a></i>;<br/>
<a NAME="E20:9"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <font color="Maroon">m</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E21:9"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>


</div>
