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

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



<b>assume </b><a NAME="E1:28"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="int_2.html#R2">are_relative_prime</a>  )
 ;<br/>

<a NAME="E2:28"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
;<br/>
<a NAME="E3:28"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="nat_1.html#T40">NAT_1:40</a></i>;<br/>
<a NAME="E4:28"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E5:28"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E6:28"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">C</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> ) </span>}</span> ;<br/>
<a NAME="E7:28"/><i><font color="Green">E47</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:28_1"/>
<font color="Maroon">n</font>,<font color="Maroon">i</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E5:28_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 ;<br/>
<b>thus </b><a NAME="E6:28_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E3:28"><i><font color="Green">E34</font></i></a>, <a class="txt" href="euler_1.html#E4"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">n</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E9:28"/><i><font color="Green">E63</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:28_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:28_2"/>
<font color="Maroon">m</font>,<font color="Maroon">i</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E5:28_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 ;<br/>
<b>thus </b><a NAME="E6:28_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E4"><i><font color="Green">Lemma35</font></i></a>, <a class="txt" href="euler_1.html#E5"><i><font color="Green">Lemma36</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E11:28"/><i><font color="Green">E75</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:28_3"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E4:28_3"/><i><font color="Green">E89</font></i>: 
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">y</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E6:28_3"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E7:28_3"/>
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E8:28_3"/>
( <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="euler_1.html#T1" target="_self">Th1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_3_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_3_1_1"><b>suppose </b></a><a NAME="E1:28_3_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:28_3_1_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:28_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">i1</font> = <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">i2</font> = <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>;<br/>
<i><font color="Green">E92</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</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">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green">E30</font></i></a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D7">INT_1:def 7</a></i>
;<br/>
<a NAME="E2:28_3_1_1_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#T52">INT_1:52</a></i>;<br/>
<b>hence </b><a NAME="E3:28_3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green">E30</font></i></a>, <a class="ref" href="euler_1.html#T3" target="_self">Th3</a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E4:28_3_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="euler_1.html#E5"><i><font color="Green">Lemma36</font></i></a>, <a class="txt" href="euler_1.html#E7"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="euler_1.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E5:28_3_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_3_1_2"><b>suppose </b></a><a NAME="E1:28_3_1_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:28_3_1_2"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:28_3_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="euler_1.html#E5"><i><font color="Green">Lemma36</font></i></a>, <a class="ref" href="euler_1.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">C</font> = <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E13:28"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:28_4"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_4_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:28_4_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E4:28_4_1"/><i><font color="Green">E102</font></i>: 
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">y</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E6:28_4_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E7:28_4_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E8:28_4_1"/>
( <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="ref" href="euler_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E9:28_4_1"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_4_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">i1</font> = <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">i2</font> = <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>;<br/>
<i><font color="Green">E106</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</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">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green">E30</font></i></a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D7">INT_1:def 7</a></i>
;<br/>
<a NAME="E2:28_4_1_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#T52">INT_1:52</a></i>;<br/>
<b>hence </b><a NAME="E3:28_4_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green">E30</font></i></a>, <a class="ref" href="euler_1.html#T4" target="_self">Th4</a>, </i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_4_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_4_1_2"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_4_1_2_1"><b>suppose </b></a><a NAME="E1:28_4_1_2_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:28_4_1_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>hence </b><a NAME="E3:28_4_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="euler_1.html#E5"><i><font color="Green">Lemma36</font></i></a>, <a class="txt" href="euler_1.html#E7"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="euler_1.html#T2" target="_self">Th2</a>, <a class="ref" href="euler_1.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_4_1_2_2"><b>suppose </b></a><a NAME="E1:28_4_1_2_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:28_4_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K10">hcf</a> 0 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="euler_1.html#T2" target="_self">Th2</a>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/><b>hence </b><a NAME="E3:28_4_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="euler_1.html#E3"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="newton.html#T65">NEWTON:65</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:28_4"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Olive">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:28_4"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E5:28_4"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E6:28_4"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 ;<br/>
<a NAME="E7:28_4"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_4_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_4_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:28_4_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="euler_1.html#T1" target="_self">Th1</a>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E3:28_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_1.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="nat_1.html#T40">NAT_1:40</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:28_4"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E7"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">y0</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">x0</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E10:28_4"/><i><font color="Green">E113</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E1"><i><font color="Green">Lemma27</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">y1</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E12:28_4"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E13:28_4"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y0</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
 ;<br/>
<a NAME="E14:28_4"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">x0</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon">j</font> = <font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font>;<br/>
<i><font color="Green">E119</font></i>: <font color="Maroon">x1</font> = 
<font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="euler_1.html#T4" target="_self">Th4</a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>
<br/>.= 
<font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">m</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D7">INT_1:def 7</a></i>
;<br/>
<a NAME="E16:28_4"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">m</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T52">INT_1:52</a></i>;<br/>
<a NAME="E17:28_4"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="txt" href="euler_1.html#E4:28"><i><font color="Green">E44</font></i></a>, </i>;<br/>
<a NAME="E18:28_4"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">y0</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y0</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon">k</font> = <font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font>;<br/>
<i><font color="Green">E122</font></i>: <font color="Maroon">y1</font> = 
<font color="Maroon">y0</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, , <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>
<br/>.= 
<font color="Maroon">y0</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D7">INT_1:def 7</a></i>
;<br/>
<a NAME="E20:28_4"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y0</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T52">INT_1:52</a></i>;<br/>
<a NAME="E21:28_4"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="txt" href="euler_1.html#E7:28"><i><font color="Green">E47</font></i></a>, </i>;<br/>
<a NAME="E22:28_4"/><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euler_1.html#T4" target="_self">Th4</a>, , , <a class="txt" href="euler_1.html#E6:28"><i><font color="Green">E46</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</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">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green">E30</font></i></a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euler_1.html#T2" target="_self">Th2</a>, <a class="ref" href="euler_1.html#T3" target="_self">Th3</a>, <a class="ref" href="pre_ff.html#T4">PRE_FF:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span>

;<br/>
<a NAME="E24:28_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E25:28_4"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E5"><i><font color="Green">Lemma36</font></i></a>, <a class="ref" href="euler_1.html#T1" target="_self">Th1</a>, <a class="ref" href="euler_1.html#T3" target="_self">Th3</a>, <a class="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, <a class="ref" href="euler_1.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:28"/><b>then </b><i><font color="Green">E125</font></i>: 
 <a href="euler_1.html#K1" target="_self">Euler</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">C</font>
 
;<br/>
<a NAME="E15:28"/><i><font color="Green">E126</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 1 holds <br/><font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_5"><b>proof </b></a><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>assume </b><a NAME="E1:28_5"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E2:28_5"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T64">NEWTON:64</a></i>;<br/>
<a NAME="E3:28_5"/><b>then </b>
<font color="Maroon">n</font>,<font color="Maroon">a</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:28_5"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</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#E7"><i><font color="Green">Lemma38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:28"/><i><font color="Green">E128</font></i>: 
for <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">b</font> <a href="hidden.html#R1">=</a> 1 holds <br/><font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_6"><b>proof </b></a><div class="add">

<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>assume </b><a NAME="E1:28_6"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E2:28_6"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T64">NEWTON:64</a></i>;<br/>
<a NAME="E3:28_6"/><b>then </b>
<font color="Maroon">m</font>,<font color="Maroon">b</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:28_6"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</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="ref" href="euler_1.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:28"/><i><font color="Green">E130</font></i>: 
for <font color="Olive">y</font>, <font color="Olive">x</font> being  <a href="int_1.html#NM1">Integer</a>  st <font color="Olive">y</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> 1 holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>



<b>assume </b><a NAME="E1:28_7"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/>

<b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E3:28_7"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">y'</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">x'</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="euler_1.html#T3" target="_self">Th3</a>, <a class="ref" href="nat_1.html#T23">NAT_1:23</a></i>;<br/>
<a NAME="E4:28_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">y'</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">x'</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E5:28_7"/><b>then </b><i><font color="Green">E134</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">y'</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">x'</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:28_7"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>
<a NAME="E8:28_7"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E9:28_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E10:28_7"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">n</font>,<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E11:28_7"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:28_7"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">m</font>,<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E13:28_7"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="nat_1.html#T23">NAT_1:23</a></i>;<br/>
<a NAME="E14:28_7"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="txt" href="euler_1.html#E4:28"><i><font color="Green">E44</font></i></a>, <a class="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, </i>;<br/>
<a NAME="E15:28_7"/><b>then </b><i><font color="Green">E139</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">d</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E17:28_7"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">d</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E18:28_7"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green">E30</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<a NAME="E19:28_7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E6:28"><i><font color="Green">E46</font></i></a>, <a class="txt" href="euler_1.html#E7:28"><i><font color="Green">E47</font></i></a>, </i>;<br/>
<a NAME="E20:28_7"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Maroon">t</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="euler_1.html#T3" target="_self">Th3</a>, <a class="ref" href="euler_1.html#T4" target="_self">Th4</a>, , <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E21:28_7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="euler_1.html#T3" target="_self">Th3</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:28"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E19:28"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font>, <font color="Maroon">B</font> = <font color="Maroon">B</font>, <font color="Maroon">C</font> = <font color="Maroon">C</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="txt" href="euler_1.html#E3:28"><i><font color="Green">E34</font></i></a>, <a class="txt" href="euler_1.html#E4"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="euler_1.html#T1" target="_self">Th1</a>, <a class="txt" href="euler_1.html#E7"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="euler_1.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>,   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>;<br/>
<a NAME="E21:28"/><i><font color="Green">E143</font></i>: 
for <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font><br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">y</font>,<font color="Olive">x</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font>;<br/>



<a NAME="E1:28_8"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
;<br/>
<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:28_8"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E4:28_8"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">n</font>,<font color="Maroon">i</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E5:28_8"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>and </b><br/><a NAME="E6:28_8"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>
<a NAME="E7:28_8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:28_8"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E10:28_8"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">m</font>,<font color="Maroon">j</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E11:28_8"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>and </b><br/><a NAME="E12:28_8"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>
<a NAME="E13:28_8"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i>, <a class="txt" href="euler_1.html#E7:28"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">ii</font> = <font color="Maroon">i</font>, <font color="Maroon">jj</font> = <font color="Maroon">j</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E15:28_8"/>
( <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font> )
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, , <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>
<a NAME="E16:28_8"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T23">NAT_1:23</a></i>;<br/>
<a NAME="E17:28_8"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E18:28_8"/><b>then </b><i><font color="Green">E155</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">ii</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">jj</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">q</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E20:28_8"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">ii</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">jj</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>
<a NAME="E21:28_8"/>
<font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E22:28_8"/><b>then </b>
<font color="Maroon">i</font>,<font color="Maroon">n</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E23:28_8"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E1"><i><font color="Green">Lemma27</font></i></a>, , </i>;<br/>
<a NAME="E24:28_8"/><b>then </b><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E25:28_8"/>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E6:28"><i><font color="Green">E46</font></i></a>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E26:28_8"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">j</font>,<font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E27:28_8"/>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E28:28_8"/><b>then </b>
<font color="Maroon">n</font>,<font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E29:28_8"/><b>then </b>
<font color="Maroon">j</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, , <a class="ref" href="euler_1.html#T7" target="_self">Th7</a>, </i>;<br/>
<a NAME="E30:28_8"/><b>then </b><i><font color="Green">E160</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E31:28_8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E32:28_8"/><b>then </b><i><font color="Green">E161</font></i>: 
<font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E33:28_8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E34:28_8"/><b>then </b><i><font color="Green">E162</font></i>: 
<font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E35:28_8"/>
( <font color="Maroon">i</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, , <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>
<a NAME="E36:28_8"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T23">NAT_1:23</a></i>;<br/>
<a NAME="E37:28_8"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, , <a class="ref" href="euler_1.html#T8" target="_self">Th8</a>, </i>;<br/>
<a NAME="E38:28_8"/><b>then </b><i><font color="Green">E163</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E40:28_8"/><i><font color="Green">E164</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E41:28_8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green">E30</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">l</font> = <font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>;<br/>
<a NAME="E42:28_8"/>
<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#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="euler_1.html#T9" target="_self">Th9</a>, , </i>;<br/>
<a NAME="E43:28_8"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font>,<font color="Maroon">q</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="euler_1.html#T6" target="_self">Th6</a>, , <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E44:28_8"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">y</font>,<font color="Maroon">x</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="euler_1.html#T4" target="_self">Th4</a>, <a class="txt" href="euler_1.html#E4:28"><i><font color="Green">E44</font></i></a>, <a class="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, <a class="ref" href="euler_1.html#T5" target="_self">Th5</a>, <a class="ref" href="euler_1.html#T6" target="_self">Th6</a>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">C</font><b> such that </b><br/><a NAME="E23:28"/><i><font color="Green">E165</font></i>: 
for <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font><br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">y</font>,<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_7.html#S1">FUNCT_7:sch 1</a>(<a class="ref" href="euler_1.html#T3" target="_self">Th3</a>);<br/></i>
<a NAME="E24:28"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<a NAME="E1:28_9"/><i><font color="Green">E167</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>assume </b><a NAME="E2:28_9"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">xx1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">xx2</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E4:28_9"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">xx1</font>,<font color="Maroon">xx2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E5:28_9"/>
<font color="Maroon">xx1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
;<br/>
<b>then consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:28_9"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">xx1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 <b>and </b><br/><a NAME="E8:28_9"/>
<font color="Maroon">n</font>,<font color="Maroon">x1</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E9:28_9"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">x1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>and </b><br/><a NAME="E10:28_9"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>
<a NAME="E11:28_9"/>
<font color="Maroon">xx2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
;<br/>
<b>then consider </b><font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E13:28_9"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">xx2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>and </b><br/><a NAME="E14:28_9"/>
<font color="Maroon">m</font>,<font color="Maroon">x2</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E15:28_9"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">x2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>and </b><br/><a NAME="E16:28_9"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>
<b>assume </b><a NAME="E17:28_9"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">yy1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">yy2</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E19:28_9"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">yy1</font>,<font color="Maroon">yy2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E20:28_9"/>
<font color="Maroon">yy1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
;<br/>
<b>then consider </b><font color="Maroon">y1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E22:28_9"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">yy1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 <b>and </b><br/><a NAME="E23:28_9"/>
<font color="Maroon">n</font>,<font color="Maroon">y1</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E24:28_9"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">y1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>and </b><br/><a NAME="E25:28_9"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>
<a NAME="E26:28_9"/>
<font color="Maroon">yy2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
;<br/>
<b>then consider </b><font color="Maroon">y2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E28:28_9"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">yy2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 <b>and </b><br/><a NAME="E29:28_9"/>
<font color="Maroon">m</font>,<font color="Maroon">y2</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E30:28_9"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">y2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>and </b><br/><a NAME="E31:28_9"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">y2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>
<b>assume </b><a NAME="E32:28_9"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>

<i><font color="Green">E189</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="euler_1.html#T4" target="_self">Th4</a>, <a class="txt" href="euler_1.html#E4:28"><i><font color="Green">E44</font></i></a>, <a class="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, <a class="ref" href="euler_1.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">y2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="euler_1.html#T4" target="_self">Th4</a>, , , , </i>
;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y1</font>, <font color="Maroon">y2</font> = <font color="Maroon">y2</font>, <font color="Maroon">x1</font> = <font color="Maroon">x1</font>, <font color="Maroon">x2</font> = <font color="Maroon">x2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>;<br/>
<a NAME="E35:28_9"/><i><font color="Green">E190</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">y1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">y2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</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">x1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</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">y1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">y2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/>
<a NAME="E36:28_9"/><i><font color="Green">E191</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">y2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E37:28_9"/><i><font color="Green">E192</font></i>: 
<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">x1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">x1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">x2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</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">x1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">y1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><font color="Maroon">y2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</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">x1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">y2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 
;<br/>
<b>set </b><font color="Maroon">w</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">y2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>;<br/>
<a NAME="E38:28_9"/>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E39:28_9"/><b>then </b><i><font color="Green">E194</font></i>: 
<font color="Maroon">n</font>,<font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E41:28_9"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/>
<a NAME="E42:28_9"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">n</font>,<font color="Maroon">p</font> <a href="int_2.html#R1">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E9:28"><i><font color="Green">E63</font></i></a>, , </i>;<br/>
<a NAME="E43:28_9"/><i><font color="Green">E198</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</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">x1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">x2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">y1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">y2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , , , <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/>
<a NAME="E44:28_9"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">x2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">y2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E45:28_9"/><b>then </b>
<font color="Maroon">p</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E46:28_9"/><b>then </b><i><font color="Green">E199</font></i>: 
<font color="Maroon">m</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y2</font>
 
<b>by </b><i>, , <a class="ref" href="int_2.html#T40">INT_2:40</a></i>;<br/>
<a NAME="E47:28_9"/>
<font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_9_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_9_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_9_2_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y2</font> or <font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y2</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_9_2_1_1"><b>suppose </b></a><a NAME="E1:28_9_2_1_1"/><i><font color="Green">E200</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y2</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">k</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:28_9_2_1_1"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y2</font>
 ;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</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="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E5:28_9_2_1_1"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="euler_1.html#T10" target="_self">Th10</a>, , </i>;<br/><a NAME="E6:28_9_2_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_9_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_9_2_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:28_9_2_1_1_1"/><b>then </b><i><font color="Green">E203</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
;<br/>
<a NAME="E3:28_9_2_1_1_1"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">y2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> 1
 
<b>by </b><i><a class="ref" href="euler_1.html#T8" target="_self">Th8</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E4:28_9_2_1_1_1"/><b>then </b><i><font color="Green">E204</font></i>: 
<font color="Maroon">x2</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">y2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="euler_1.html#T6" target="_self">Th6</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>set </b><font color="Maroon">b</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>;<br/>
<a NAME="E5:28_9_2_1_1_1"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> is    <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="E6:28_9_2_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E7:28_9_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="euler_1.html#T11" target="_self">Th11</a>, , <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:28_9_2_1_1"/>
<font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_9_2_1_2"><b>suppose </b></a><a NAME="E1:28_9_2_1_2"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y2</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">k</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:28_9_2_1_2"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y2</font></span>)</span>
 ;<br/><a NAME="E4:28_9_2_1_2"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">m</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="euler_1.html#T10" target="_self">Th10</a>, , <a class="ref" href="int_2.html#T14">INT_2:14</a></i>;<br/><a NAME="E5:28_9_2_1_2"/><i><font color="Green">E208</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</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="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E7:28_9_2_1_2"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="euler_1.html#T11" target="_self">Th11</a>, </i>;<br/><a NAME="E8:28_9_2_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_9_2_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_9_2_1_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:28_9_2_1_2_1"/><b>then </b><i><font color="Green">E210</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
;<br/>
<a NAME="E3:28_9_2_1_2_1"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E4:28_9_2_1_2_1"/><b>then </b><i><font color="Green">E211</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="euler_1.html#T9" target="_self">Th9</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>set </b><font color="Maroon">b</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>;<br/>
<a NAME="E5:28_9_2_1_2_1"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> is    <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="E6:28_9_2_1_2_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E7:28_9_2_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:28_9_2_1_2"/><b>then </b>
0 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y2</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:28_9_2_1_2"/>
<font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> 0
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E48:28_9"/><b>then </b><i><font color="Green">E212</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 
;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E50:28_9"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/>
<a NAME="E51:28_9"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">m</font>,<font color="Maroon">p</font> <a href="int_2.html#R1">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E1"><i><font color="Green">Lemma27</font></i></a>, , </i>;<br/>
<a NAME="E52:28_9"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</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="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">x2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">y1</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><font color="Maroon">y2</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y2</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E53:28_9"/><b>then </b>
<font color="Maroon">p</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E54:28_9"/><b>then </b><i><font color="Green">E215</font></i>: 
<font color="Maroon">n</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font>
 
<b>by </b><i>, <a class="ref" href="euler_1.html#T11" target="_self">Th11</a>, <a class="ref" href="int_2.html#T40">INT_2:40</a></i>;<br/>
<a NAME="E55:28_9"/>
<font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_9_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_9_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_9_3_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font> or <font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_9_3_1_1"><b>suppose </b></a><a NAME="E1:28_9_3_1_1"/><i><font color="Green">E216</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">k</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:28_9_3_1_1"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font>
 ;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</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="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E5:28_9_3_1_1"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, , , </i>;<br/><a NAME="E6:28_9_3_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_9_3_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_9_3_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:28_9_3_1_1_1"/><b>then </b><i><font color="Green">E219</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
;<br/>
<a NAME="E3:28_9_3_1_1_1"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> 1
 
<b>by </b><i><a class="ref" href="euler_1.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E4:28_9_3_1_1_1"/><b>then </b><i><font color="Green">E220</font></i>: 
<font color="Maroon">x1</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">y1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E7:28"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>set </b><font color="Maroon">b</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>;<br/>
<a NAME="E5:28_9_3_1_1_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> is    <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="E6:28_9_3_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E7:28_9_3_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="euler_1.html#E11:28"><i><font color="Green">E75</font></i></a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:28_9_3_1_1"/>
<font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_9_3_1_2"><b>suppose </b></a><a NAME="E1:28_9_3_1_2"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">k</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:28_9_3_1_2"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font></span>)</span>
 ;<br/><a NAME="E4:28_9_3_1_2"/><i><font color="Green">E223</font></i>: 
<font color="Maroon">n</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="int_2.html#T14">INT_2:14</a></i>;<br/><a NAME="E5:28_9_3_1_2"/><i><font color="Green">E224</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</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="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E7:28_9_3_1_2"/><i><font color="Green">E225</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, , </i>;<br/><a NAME="E8:28_9_3_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_9_3_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_9_3_1_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:28_9_3_1_2_1"/><b>then </b><i><font color="Green">E226</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
;<br/>
<a NAME="E3:28_9_3_1_2_1"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E6:28"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E4:28_9_3_1_2_1"/><b>then </b><i><font color="Green">E227</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">x1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>set </b><font color="Maroon">b</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>;<br/>
<a NAME="E5:28_9_3_1_2_1"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i>, <a class="txt" href="euler_1.html#E11:28"><i><font color="Green">E75</font></i></a>, , <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E6:28_9_3_1_2_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E7:28_9_3_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="euler_1.html#T12" target="_self">Th12</a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:28_9_3_1_2"/><b>then </b>
0 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:28_9_3_1_2"/>
<font color="Maroon">x1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E56:28_9"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E4:28"><i><font color="Green">E44</font></i></a>, <a class="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, <a class="ref" href="euler_1.html#T5" target="_self">Th5</a>, , , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:28"/>
<font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:28_10"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:28_10"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:28_10"/><i><font color="Green">E228</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E5:28_10"/><i><font color="Green">E229</font></i>: 
 ex <font color="Olive">x0</font>, <font color="Olive">y0</font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">y0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Olive">x0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/>
<b>consider </b><font color="Maroon">x0</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">y0</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E7:28_10"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E8:28_10"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E9:28_10"/>
( <font color="Maroon">y0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">x0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="euler_1.html#E4:28"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">y</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E11:28_10"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E12:28_10"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y0</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
 ;<br/>
<a NAME="E13:28_10"/><i><font color="Green">E234</font></i>: 
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="txt" href="euler_1.html#E7:28"><i><font color="Green">E47</font></i></a>, <a class="ref" href="euler_1.html#T5" target="_self">Th5</a>, <a class="ref" href="newton.html#T79">NEWTON:79</a></i>;<br/>
<a NAME="E14:28_10"/><i><font color="Green">E235</font></i>: 
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="txt" href="euler_1.html#E7:28"><i><font color="Green">E47</font></i></a>, <a class="ref" href="euler_1.html#T5" target="_self">Th5</a>, <a class="ref" href="newton.html#T78">NEWTON:78</a></i>;<br/>
<a NAME="E15:28_10"/><i><font color="Green">E236</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="euler_1.html#T5" target="_self">Th5</a>, <a class="ref" href="newton.html#T78">NEWTON:78</a></i>;<br/>
<a NAME="E16:28_10"/><i><font color="Green">E237</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_10_1"/><i><font color="Green">E238</font></i>: 
<font color="Maroon">x0</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon">j</font> = <font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font>;<br/>
<a NAME="E2:28_10_1"/><i><font color="Green">E239</font></i>: 
<font color="Maroon">y0</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y0</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon">k</font> = <font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font>;<br/>
<a NAME="E3:28_10_1"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E7:28"><i><font color="Green">E47</font></i></a>, <a class="ref" href="euler_1.html#T5" target="_self">Th5</a>, <a class="ref" href="euler_1.html#T7" target="_self">Th7</a>, </i>;<br/>
<b>hence </b><a NAME="E4:28_10_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E17:28_10"/><i><font color="Green">E240</font></i>: 
<font color="Maroon">m</font>,<font color="Maroon">x0</font> <a href="int_2.html#R1">are_relative_prime</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">p</font> = <font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font>;<br/>
<a NAME="E1:28_10_2"/><i><font color="Green">E241</font></i>: 
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font> is    <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_2.html#T29">INT_2:29</a></i>;<br/>
<a NAME="E2:28_10_2"/>
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T31">INT_2:31</a></i>;<br/>
<b>then consider </b><font color="Maroon">i1</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:28_10_2"/><i><font color="Green">E242</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="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E5:28_10_2"/>
( <font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">i1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="euler_1.html#E1"><i><font color="Green">Lemma27</font></i></a>, </i>;<br/>
<a NAME="E6:28_10_2"/><b>then </b><i><font color="Green">E243</font></i>: 
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E7:28_10_2"/>
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">x0</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T32">INT_2:32</a></i>;<br/>
<b>then consider </b><font color="Maroon">i2</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E9:28_10_2"/><i><font color="Green">E244</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E10:28_10_2"/><i><font color="Green">E245</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E6:28"><i><font color="Green">E46</font></i></a>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E11:28_10_2"/>
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E12:28_10_2"/><b>then </b><i><font color="Green">E246</font></i>: 
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></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">i</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E13:28_10_2"/><i><font color="Green">E247</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</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"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</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="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <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#E1:28"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="euler_1.html#T9" target="_self">Th9</a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon">k</font> = <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">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</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="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</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="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>;<br/>
<a NAME="E14:28_10_2"/><i><font color="Green">E248</font></i>: 
<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">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</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="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</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"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><font color="Maroon">n</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="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
;<br/>
<b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>;<br/>
<a NAME="E15:28_10_2"/>
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_10_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<a NAME="E2:28_10_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><font color="Maroon">n</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="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><font color="Maroon">n</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="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euler_1.html#T8" target="_self">Th8</a>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<b>hence </b><a NAME="E3:28_10_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green">E30</font></i></a>, <a class="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="euler_1.html#T9" target="_self">Th9</a>, , <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font>, <font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <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>, <a class="ref" href="int_2.html#T29">INT_2:29</a></i>;<br/>
<a NAME="E17:28_10_2"/><i><font color="Green">E249</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">jj</font> = <font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font>;<br/>
<a NAME="E18:28_10_2"/>
not <font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_10_2_2"/>
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<a NAME="E2:28_10_2_2"/><b>then </b>
<font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euler_1.html#T8" target="_self">Th8</a>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<b>hence </b><a NAME="E3:28_10_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green">E31</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">jj</font> = <font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</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="int_1.html#T16">INT_1:16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_10_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_10_2_3_1"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">jj</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) or ( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">jj</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 ) )
 </span><b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, , <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_10_2_3_1_1"><b>suppose </b></a><a NAME="E1:28_10_2_3_1_1"/><i><font color="Green">E250</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">jj</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:28_10_2_3_1_1"/><b>then </b><i><font color="Green">E251</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:28_10_2_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">jj</font></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, , , , <a class="ref" href="int_2.html#D3">INT_2:def 3</a></i>;<br/><a NAME="E4:28_10_2_3_1_1"/><b>then </b><i><font color="Green">E252</font></i>: 
<font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">jj</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="euler_1.html#E9:28"><i><font color="Green">E63</font></i></a>, , </i>;<br/><a NAME="E5:28_10_2_3_1_1"/>
<font color="Maroon">jj</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">j</font> is    <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_2.html#T29">INT_2:29</a></i>;<br/><b>hence </b><a NAME="E6:28_10_2_3_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T40">NAT_1:40</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_10_2_3_1_2"><b>suppose </b></a><a NAME="E1:28_10_2_3_1_2"/>
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">jj</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:28_10_2_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">jj</font></span>)</span> <a href="nat_1.html#K10">hcf</a> 0 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, <a class="txt" href="euler_1.html#E6:28"><i><font color="Green">E46</font></i></a>, , , , <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/><a NAME="E3:28_10_2_3_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">jj</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T65">NEWTON:65</a></i>;<br/><b>hence </b><a NAME="E4:28_10_2_3_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T40">NAT_1:40</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E21:28_10_2"/>
<font color="Maroon">m</font>,<font color="Maroon">x0</font> <a href="int_2.html#R1">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D4">INT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:28_10"/><i><font color="Green">E253</font></i>: 
<font color="Maroon">n</font>,<font color="Maroon">y0</font> <a href="int_2.html#R1">are_relative_prime</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">p</font> = <font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font>;<br/>
<a NAME="E1:28_10_3"/><i><font color="Green">E254</font></i>: 
<font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font> is    <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_2.html#T29">INT_2:29</a></i>;<br/>
<a NAME="E2:28_10_3"/>
<font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T31">INT_2:31</a></i>;<br/>
<b>then consider </b><font color="Maroon">i1</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E4:28_10_3"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E5:28_10_3"/>
( <font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">i1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="euler_1.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="euler_1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E6:28_10_3"/><b>then </b><i><font color="Green">E256</font></i>: 
<font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E7:28_10_3"/>
<font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">y0</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T32">INT_2:32</a></i>;<br/>
<b>then consider </b><font color="Maroon">i2</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E9:28_10_3"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">y0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<a NAME="E10:28_10_3"/><i><font color="Green">E258</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E6:28"><i><font color="Green">E46</font></i></a>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E11:28_10_3"/>
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E12:28_10_3"/><b>then </b><i><font color="Green">E259</font></i>: 
<span class="p1">(<span class="default"><a href="int_2.html#K1">abs</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></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">i</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E13:28_10_3"/><i><font color="Green">E260</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</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">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green">E30</font></i></a>, <a class="ref" href="euler_1.html#T8" target="_self">Th8</a>, , <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span></span>)</span>;<br/>
<a NAME="E14:28_10_3"/><i><font color="Green">E261</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</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">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</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="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
;<br/>
<b>set </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span>;<br/>
<a NAME="E15:28_10_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_10_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<a NAME="E2:28_10_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</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="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K3">*</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="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euler_1.html#T9" target="_self">Th9</a>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<b>hence </b><a NAME="E3:28_10_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green">E30</font></i></a>, <a class="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, <a class="ref" href="euler_1.html#T8" target="_self">Th8</a>, , , <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font>, <font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</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>, <a class="ref" href="int_2.html#T29">INT_2:29</a></i>;<br/>
<a NAME="E17:28_10_3"/><i><font color="Green">E262</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euler_1.html#T8" target="_self">Th8</a></i>;<br/>
<b>set </b><font color="Maroon">jj</font> = <font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font>;<br/>
<a NAME="E18:28_10_3"/>
not <font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_10_3_2"/>
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<a NAME="E2:28_10_3_2"/><b>then </b>
<font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="euler_1.html#T9" target="_self">Th9</a>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<b>hence </b><a NAME="E3:28_10_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green">E31</font></i></a>, <a class="ref" href="euler_1.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">jj</font> = <font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</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="int_1.html#T16">INT_1:16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10_3_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_10_3_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_10_3_3_1"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">jj</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) or ( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">jj</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 ) )
 </span><b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="txt" href="euler_1.html#E9:28"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_10_3_3_1_1"><b>suppose </b></a><a NAME="E1:28_10_3_3_1_1"/><i><font color="Green">E263</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">jj</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:28_10_3_3_1_1"/><b>then </b><i><font color="Green">E264</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:28_10_3_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">jj</font></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, <a class="ref" href="euler_1.html#T8" target="_self">Th8</a>, , , <a class="ref" href="int_2.html#D3">INT_2:def 3</a></i>;<br/><a NAME="E4:28_10_3_3_1_1"/><b>then </b><i><font color="Green">E265</font></i>: 
<font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">jj</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E5:28_10_3_3_1_1"/>
<font color="Maroon">jj</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">j</font> is    <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_2.html#T29">INT_2:29</a></i>;<br/><b>hence </b><a NAME="E6:28_10_3_3_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T40">NAT_1:40</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_10_3_3_1_2"><b>suppose </b></a><a NAME="E1:28_10_3_3_1_2"/>
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">jj</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:28_10_3_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">jj</font></span>)</span> <a href="nat_1.html#K10">hcf</a> 0 <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, <a class="txt" href="euler_1.html#E6:28"><i><font color="Green">E46</font></i></a>, <a class="ref" href="euler_1.html#T8" target="_self">Th8</a>, , , <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/><a NAME="E3:28_10_3_3_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">jj</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T65">NEWTON:65</a></i>;<br/><b>hence </b><a NAME="E4:28_10_3_3_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T40">NAT_1:40</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E21:28_10_3"/>
<font color="Maroon">n</font>,<font color="Maroon">y0</font> <a href="int_2.html#R1">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D4">INT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</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="euler_1.html#T6" target="_self">Th6</a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E20:28_10"/><i><font color="Green">E266</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_10_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:28_10_4"/><b>then </b><i><font color="Green">E267</font></i>: 
<font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="txt" href="euler_1.html#E7:28"><i><font color="Green">E47</font></i></a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon">j</font> = <font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font>;<br/>
<i><font color="Green">E268</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</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="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euler_1.html#T8" target="_self">Th8</a>, <a class="txt" href="euler_1.html#E1:28"><i><font color="Green">E30</font></i></a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>
<br/>.= 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">n</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="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<b>set </b><font color="Maroon">jj</font> = <span class="p1">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</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="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">n</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="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></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="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, <a class="ref" href="euler_1.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E5:28_10_4"/><i><font color="Green">E270</font></i>: 
( <font color="Maroon">g</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> 0 )
 
;<br/>
<a NAME="E6:28_10_4"/><i><font color="Green">E271</font></i>: 
not <span class="p1">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</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="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10_4_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_10_4_2"/>
<span class="p1">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</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="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<a NAME="E2:28_10_4_2"/><b>then </b>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">n</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="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">n</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="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</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>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<b>hence </b><a NAME="E3:28_10_4_2"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:28_10_4"/><i><font color="Green">E272</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E6:28"><i><font color="Green">E46</font></i></a>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E8:28_10_4"/>
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E9:28_10_4"/><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">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></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">i</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E10:28_10_4"/><b>then </b><i><font color="Green">E273</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, <a class="ref" href="euler_1.html#T9" target="_self">Th9</a>, <a class="ref" href="int_2.html#D3">INT_2:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">jj</font> = <span class="p1">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</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="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">y0</font> <a href="xcmplx_0.html#K2">+</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></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="E12:28_10_4"/>
<font color="Maroon">jj</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10_4_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_10_4_3"/>
not <font color="Maroon">jj</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:28_10_4_3"/><b>then </b><i><font color="Green">E274</font></i>: 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">jj</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, , </i>;<br/>
<a NAME="E3:28_10_4_3"/>
<font color="Maroon">n</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">jj</font> is    <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_2.html#T29">INT_2:29</a></i>;<br/>
<b>hence </b><a NAME="E4:28_10_4_3"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_1.html#E1"><i><font color="Green">Lemma27</font></i></a>, , <a class="ref" href="nat_1.html#T40">NAT_1:40</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:28_10_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K10">hcf</a> 0 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, <a class="txt" href="euler_1.html#E6:28"><i><font color="Green">E46</font></i></a>, <a class="ref" href="euler_1.html#T9" target="_self">Th9</a>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E14:28_10_4"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_1.html#E3"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="newton.html#T65">NEWTON:65</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:28_10"/><b>then </b><i><font color="Green">E275</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="euler_1.html#T6" target="_self">Th6</a>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/>
<a NAME="E22:28_10"/><i><font color="Green">E276</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_10_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:28_10_5"/><b>then </b><i><font color="Green">E277</font></i>: 
<font color="Maroon">y0</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="euler_1.html#T5" target="_self">Th5</a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon">j</font> = <font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font>;<br/>
<i><font color="Green">E278</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">n</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="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="euler_1.html#E1:28"><i><font color="Green">E30</font></i></a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</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="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<b>set </b><font color="Maroon">jj</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</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="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></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="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, </i>;<br/>
<a NAME="E5:28_10_5"/><i><font color="Green">E279</font></i>: 
( <font color="Maroon">g</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> 0 )
 
;<br/>
<a NAME="E6:28_10_5"/><i><font color="Green">E280</font></i>: 
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10_5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_10_5_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/>

<a NAME="E2:28_10_5_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</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="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="xcmplx_0.html#K3">*</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="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</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>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/>
<b>hence </b><a NAME="E3:28_10_5_2"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:28_10_5"/><i><font color="Green">E281</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E6:28"><i><font color="Green">E46</font></i></a>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E8:28_10_5"/>
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="int_2.html#K1">abs</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E9:28_10_5"/><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">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></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">i</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E10:28_10_5"/><b>then </b><i><font color="Green">E282</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="int_2.html#D3">INT_2:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">jj</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></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="E12:28_10_5"/>
<font color="Maroon">jj</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10_5_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_10_5_3"/>
not <font color="Maroon">jj</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:28_10_5_3"/><b>then </b><i><font color="Green">E283</font></i>: 
<font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">jj</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="txt" href="euler_1.html#E9:28"><i><font color="Green">E63</font></i></a>, </i>;<br/>
<a NAME="E3:28_10_5_3"/>
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">jj</font> is    <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_2.html#T29">INT_2:29</a></i>;<br/>
<b>hence </b><a NAME="E4:28_10_5_3"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_1.html#E1"><i><font color="Green">Lemma27</font></i></a>, , <a class="ref" href="nat_1.html#T40">NAT_1:40</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:28_10_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K10">hcf</a> 0 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, <a class="txt" href="euler_1.html#E6:28"><i><font color="Green">E46</font></i></a>, , <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E14:28_10_5"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_1.html#E3"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="newton.html#T65">NEWTON:65</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:28_10"/><b>then </b><i><font color="Green">E284</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/>
<a NAME="E24:28_10"/><i><font color="Green">E285</font></i>: 
<font color="Maroon">m</font>,<font color="Maroon">x</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10_6"><b>proof </b></a><div class="add">

<a NAME="E1:28_10_6"/><i><font color="Green">E286</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="txt" href="euler_1.html#E7:28"><i><font color="Green">E47</font></i></a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/>
<a NAME="E2:28_10_6"/>
<font color="Maroon">x0</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="int_2.html#K3">gcd</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:28_10_6"/><b>then </b><i><font color="Green">E287</font></i>: 
<font color="Maroon">x0</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, , </i>;<br/>
<a NAME="E4:28_10_6"/>
<font color="Maroon">x0</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="euler_1.html#T7" target="_self">Th7</a>, <a class="ref" href="int_2.html#D4">INT_2:def 4</a></i>;<br/>
<a NAME="E5:28_10_6"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:28_10_6"/>
<font color="Maroon">m</font>,<font color="Maroon">x</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:28_10"/><i><font color="Green">E288</font></i>: 
<font color="Maroon">n</font>,<font color="Maroon">y</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_10_7"><b>proof </b></a><div class="add">

<a NAME="E1:28_10_7"/><i><font color="Green">E289</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y0</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="euler_1.html#T5" target="_self">Th5</a>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/>
<a NAME="E2:28_10_7"/>
<font color="Maroon">y0</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">y</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y0</font> <a href="int_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:28_10_7"/><b>then </b><i><font color="Green">E290</font></i>: 
<font color="Maroon">y0</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="euler_1.html#T9" target="_self">Th9</a>, </i>;<br/>
<a NAME="E4:28_10_7"/>
<font color="Maroon">y0</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="int_2.html#D4">INT_2:def 4</a></i>;<br/>
<a NAME="E5:28_10_7"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:28_10_7"/>
<font color="Maroon">n</font>,<font color="Maroon">y</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:28_10"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="euler_1.html#T8" target="_self">Th8</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">B</font> ;<br/>
<a NAME="E28:28_10"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , </i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font> ;<br/>
<a NAME="E30:28_10"/>
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> 0
 
<b>by </b><i><a class="txt" href="euler_1.html#E2"><i><font color="Green">Lemma28</font></i></a>, , <a class="ref" href="euler_1.html#T9" target="_self">Th9</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E31:28_10"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E32:28_10"/><b>then </b><i><font color="Green">E291</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="euler_1.html#E5:28"><i><font color="Green">E45</font></i></a>, , </i>;<br/>
<a NAME="E33:28_10"/><i><font color="Green">E292</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">w</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E35:28_10"/><i><font color="Green">E293</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>
<a NAME="E36:28_10"/><i><font color="Green">E294</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="txt" href="euler_1.html#E9:28"><i><font color="Green">E63</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E37:28_10"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 
<b>by </b><i>, <a class="ref" href="euler_1.html#T4" target="_self">Th4</a>, <a class="txt" href="euler_1.html#E9:28"><i><font color="Green">E63</font></i></a></i>;<br/>
<b>hence </b><a NAME="E38:28_10"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:28"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">C</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E27:28"/>
 <a href="euler_1.html#K1" target="_self">Euler</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euler_1.html#K1" target="_self">Euler</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="euler_1.html#K1" target="_self">Euler</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="euler_1.html#E5"><i><font color="Green">Lemma36</font></i></a>, <a class="ref" href="card_2.html#T65">CARD_2:65</a></i>;<br/>


</div>
