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

<b>let </b><font color="Maroon" title="c3">m</font>, <font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  )
 ;<br/>

<a NAME="E2:28"/><b>then </b><i><font color="Green" title="E28">A2</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 
;<br/>
<a NAME="E3:28"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="nat_1.html#T15" title="NAT_1:th.15">NAT_1:15</a></i>;<br/>
<a NAME="E4:28"/><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E5:28"/><b>then </b><i><font color="Green" title="E31">A5</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E6:28"/><b>then </b><i><font color="Green" title="E32">A6</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
;<br/>
<b>set </b><font color="Maroon" title="c5">A</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c6">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c7">C</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  :  ex <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b3">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b3">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c8">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> ) </span>}</span> ;<br/>
<a NAME="E7:28"/><i><font color="Green" title="E33">A7</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">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" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green" title="E34">A8</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 <b>and </b><br/><a NAME="E4:28_1"/>
<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c10">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E5:28_1"/><i><font color="Green" title="E35">A9</font></i>: 
( <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> )
 ;<br/>
<b>thus </b><a NAME="E6:28_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E3:28_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="euler_1.html#E5:28_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">A</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> ) </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<a NAME="E9:28"/><i><font color="Green" title="E34">A10</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">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" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:28_2"/><i><font color="Green" title="E35">A11</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 <b>and </b><br/><a NAME="E4:28_2"/>
<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c11">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E5:28_2"/><i><font color="Green" title="E36">A12</font></i>: 
( <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c11">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> )
 ;<br/>
<b>thus </b><a NAME="E6:28_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E3:28_2"><i><font color="Green" title="E35">A11</font></i></a>, <a class="txt" href="euler_1.html#E5:28_2"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> ) </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<a NAME="E11:28"/><i><font color="Green" title="E35">A13</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  :  ex <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b3">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b3">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</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" title="c11">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_3"/>
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  :  ex <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b3">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b3">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:28_3"/><i><font color="Green" title="E36">A14</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z</font>
 <b>and </b><br/><a NAME="E4:28_3"/><i><font color="Green" title="E37">A15</font></i>: 
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c12">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 ;<br/>
<b>consider </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E6:28_3"/><i><font color="Green" title="E38">A16</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 <b>and </b><br/><a NAME="E7:28_3"/>
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c12">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E8:28_3"/>
( <font color="Maroon" title="c14">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c13">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="euler_1.html#E4:28_3"><i><font color="Green" title="E37">A15</font></i></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" title="c12">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c12">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:28_3_1_1"/><b>then </b><i><font color="Green" title="E39">A17</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</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" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">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" title="c15">i1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">x</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c16">i2</font> = <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>;<br/>
<i><font color="Green" title="E40">A18</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euler_1.html#E4:28"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></i>
;<br/>
<a NAME="E2:28_3_1_1_1"/>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#T52" title="INT_1:th.52">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" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E4:28"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="euler_1.html#E1:28_3_1_1_1"><i><font color="Green" title="E40">A18</font></i></a>, <a class="txt" href="euler_1.html#E5"><i><font color="Green" title="E5">Lm5</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:28_3_1_1"/><b>then </b>
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="euler_1.html#E3:28_3"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="euler_1.html#E6:28_3"><i><font color="Green" title="E38">A16</font></i></a>, <a class="txt" href="euler_1.html#E2:28_3_1_1"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E5:28_3_1_1"/>
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="E39">A19</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:28_3_1_2"/>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:28_3_1_2"/>
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="euler_1.html#E3:28_3"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="euler_1.html#E1:28_3_1_2"><i><font color="Green" title="E39">A19</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c11">C</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  :  ex <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b3">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b3">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Olive" title="b2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<a NAME="E13:28"/>
<font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">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" title="c11">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c12">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:28_4_1"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:28_4_1"/><i><font color="Green" title="E36">A20</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z</font>
 <b>and </b><br/><a NAME="E4:28_4_1"/><i><font color="Green" title="E37">A21</font></i>: 
 ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c13">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 ;<br/>
<b>consider </b><font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c15">y</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E6:28_4_1"/><i><font color="Green" title="E38">A22</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 <b>and </b><br/><a NAME="E7:28_4_1"/><i><font color="Green" title="E39">A23</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c13">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E8:28_4_1"/>
( <font color="Maroon" title="c15">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="euler_1.html#E4:28_4_1"><i><font color="Green" title="E37">A21</font></i></a></i>;<br/>
<a NAME="E9:28_4_1"/><i><font color="Green" title="E40">A24</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">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" title="c16">i1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">x</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c17">i2</font> = <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>;<br/>
<i><font color="Green" title="E41">A25</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euler_1.html#E4:28"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></i>
;<br/>
<a NAME="E2:28_4_1_1"/>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</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" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#T52" title="INT_1:th.52">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" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E4:28"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="euler_1.html#E1:28_4_1_1"><i><font color="Green" title="E41">A25</font></i></a>, <a class="txt" href="euler_1.html#E5"><i><font color="Green" title="E5">Lm5</font></i></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" title="c13">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c13">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:28_4_1_2_1"/><b>then </b>
<font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>hence </b><a NAME="E3:28_4_1_2_1"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="euler_1.html#E3:28_4_1"><i><font color="Green" title="E36">A20</font></i></a>, <a class="txt" href="euler_1.html#E6:28_4_1"><i><font color="Green" title="E38">A22</font></i></a>, <a class="txt" href="euler_1.html#E7:28_4_1"><i><font color="Green" title="E39">A23</font></i></a>, <a class="txt" href="euler_1.html#E9:28_4_1"><i><font color="Green" title="E40">A24</font></i></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" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="euler_1.html#E7:28_4_1"><i><font color="Green" title="E39">A23</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/><b>hence </b><a NAME="E3:28_4_1_2_2"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="euler_1.html#E3:28"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="newton.html#T65" title="NEWTON:th.65">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" title="c12">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:28_4"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Olive" title="b1">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:28_4"/><i><font color="Green" title="E36">A26</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z</font>
 <b>and </b><br/><a NAME="E5:28_4"/><i><font color="Green" title="E37">A27</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c13">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E6:28_4"/><i><font color="Green" title="E38">A28</font></i>: 
( <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> )
 ;<br/>
<a NAME="E7:28_4"/>
<font color="Maroon" title="c13">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">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" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>
 ;<br/>

<a NAME="E2:28_4_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E5:28_4"><i><font color="Green" title="E37">A27</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">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#E1:28"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="nat_1.html#T15" title="NAT_1:th.15">NAT_1:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:28_4"/><b>then </b><i><font color="Green" title="E39">A29</font></i>: 
<font color="Maroon" title="c13">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E6:28_4"><i><font color="Green" title="E38">A28</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">y0</font>, <font color="Maroon" title="c15">x0</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E10:28_4"/><i><font color="Green" title="E40">A30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">y0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="euler_1.html#T11" target="_self" title="EULER_1:th.11">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">x1</font>, <font color="Maroon" title="c17">y1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E12:28_4"/><i><font color="Green" title="E41">A31</font></i>: 
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x0</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">m</font>
 <b>and </b><br/><a NAME="E13:28_4"/><i><font color="Green" title="E42">A32</font></i>: 
<font color="Maroon" title="c17">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y0</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">n</font>
 ;<br/>
<a NAME="E14:28_4"/><i><font color="Green" title="E43">A33</font></i>: 
<font color="Maroon" title="c15">x0</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c15">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font>;<br/>
<i><font color="Green" title="E44">A34</font></i>: <font color="Maroon" title="c16">x1</font> = 
<font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E12:28_4"><i><font color="Green" title="E41">A31</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">m</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></i>
;<br/>
<a NAME="E16:28_4"/>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">m</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">x0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T52" title="INT_1:th.52">INT_1:52</a></i>;<br/>
<a NAME="E17:28_4"/><b>then </b><i><font color="Green" title="E45">A35</font></i>: 
<font color="Maroon" title="c16">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E15:28_4"><i><font color="Green" title="E44">A34</font></i></a>, <a class="txt" href="euler_1.html#E5"><i><font color="Green" title="E5">Lm5</font></i></a></i>;<br/>
<a NAME="E18:28_4"/><i><font color="Green" title="E46">A36</font></i>: 
<font color="Maroon" title="c14">y0</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c19">k</font> = <font color="Maroon" title="c14">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font>;<br/>
<i><font color="Green" title="E47">A37</font></i>: <font color="Maroon" title="c17">y1</font> = 
<font color="Maroon" title="c14">y0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E13:28_4"><i><font color="Green" title="E42">A32</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c14">y0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">y0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></i>
;<br/>
<a NAME="E20:28_4"/>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">y0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">n</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c14">y0</font> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T52" title="INT_1:th.52">INT_1:52</a></i>;<br/>
<a NAME="E21:28_4"/><b>then </b><i><font color="Green" title="E48">A38</font></i>: 
<font color="Maroon" title="c17">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E19:28_4"><i><font color="Green" title="E47">A37</font></i></a>, <a class="txt" href="euler_1.html#E5"><i><font color="Green" title="E5">Lm5</font></i></a></i>;<br/>
<a NAME="E22:28_4"/><i><font color="Green" title="E49">A39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">y1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">x1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E12:28_4"><i><font color="Green" title="E41">A31</font></i></a>, <a class="txt" href="euler_1.html#E13:28_4"><i><font color="Green" title="E42">A32</font></i></a>, <a class="txt" href="euler_1.html#E14:28_4"><i><font color="Green" title="E43">A33</font></i></a>, <a class="txt" href="euler_1.html#E18:28_4"><i><font color="Green" title="E46">A36</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euler_1.html#E4:28"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euler_1.html#E8:28_4"><i><font color="Green" title="E39">A29</font></i></a>, <a class="txt" href="euler_1.html#E10:28_4"><i><font color="Green" title="E40">A30</font></i></a>, <a class="ref" href="pre_ff.html#T4" title="PRE_FF:th.4">PRE_FF:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">x0</font></span>)</span>

;<br/>
<a NAME="E24:28_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c15">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">y1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">x1</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E22:28_4"><i><font color="Green" title="E49">A39</font></i></a>, <a class="ref" href="euler_1.html#T13" target="_self" title="EULER_1:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E25:28_4"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E4:28_4"><i><font color="Green" title="E36">A26</font></i></a>, <a class="txt" href="euler_1.html#E5:28_4"><i><font color="Green" title="E37">A27</font></i></a>, <a class="txt" href="euler_1.html#E10:28_4"><i><font color="Green" title="E40">A30</font></i></a>, <a class="txt" href="euler_1.html#E17:28_4"><i><font color="Green" title="E45">A35</font></i></a>, <a class="txt" href="euler_1.html#E21:28_4"><i><font color="Green" title="E48">A38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:28"/><b>then </b><i><font color="Green" title="E36">A40</font></i>: 
 <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">C</font>
 
;<br/>
<a NAME="E15:28"/><i><font color="Green" title="E37">A41</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">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" title="c12">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:28_5"/><i><font color="Green" title="E38">A42</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<a NAME="E2:28_5"/><b>then </b>
<font color="Maroon" title="c4">n</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T64" title="NEWTON:th.64">NEWTON:64</a></i>;<br/>
<a NAME="E3:28_5"/><b>then </b>
<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c12">a</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:28_5"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="euler_1.html#E1:28_5"><i><font color="Green" title="E38">A42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:28"/><i><font color="Green" title="E38">A43</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/><font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">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" title="c12">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:28_6"/><i><font color="Green" title="E39">A44</font></i>: 
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<a NAME="E2:28_6"/><b>then </b>
<font color="Maroon" title="c3">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T64" title="NEWTON:th.64">NEWTON:64</a></i>;<br/>
<a NAME="E3:28_6"/><b>then </b>
<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c12">b</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:28_6"/>
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="euler_1.html#E1:28_6"><i><font color="Green" title="E39">A44</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:28"/><i><font color="Green" title="E39">A45</font></i>: 
for <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">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" title="c12">y</font>, <font color="Maroon" title="c13">x</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>



<b>assume </b><a NAME="E1:28_7"/><i><font color="Green" title="E40">A46</font></i>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c14">y'</font> = <font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c15">x'</font> = <font color="Maroon" title="c13">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E3:28_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c14">y'</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c15">x'</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E1:28_7"><i><font color="Green" title="E40">A46</font></i></a>, <a class="ref" href="nat_1.html#T7" title="NAT_1:th.7">NAT_1:7</a></i>;<br/>
<a NAME="E4:28_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c14">y'</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c15">x'</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
;<br/>
<a NAME="E5:28_7"/><b>then </b><i><font color="Green" title="E41">A47</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c14">y'</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c15">x'</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E6:28"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="euler_1.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:28_7"/><i><font color="Green" title="E42">A48</font></i>: 
<font color="Maroon" title="c16">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 ;<br/>
<a NAME="E8:28_7"/><i><font color="Green" title="E43">A49</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E9:28_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="euler_1.html#T8" target="_self" title="EULER_1:th.8">Th8</a></i>;<br/>
<a NAME="E10:28_7"/><b>then </b><i><font color="Green" title="E44">A50</font></i>: 
<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E11:28_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E8:28_7"><i><font color="Green" title="E43">A49</font></i></a>, <a class="ref" href="euler_1.html#T8" target="_self" title="EULER_1:th.8">Th8</a></i>;<br/>
<a NAME="E12:28_7"/><b>then </b><i><font color="Green" title="E45">A51</font></i>: 
<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E13:28_7"/>
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="nat_1.html#T7" title="NAT_1:th.7">NAT_1:7</a></i>;<br/>
<a NAME="E14:28_7"/><b>then </b>
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E10:28_7"><i><font color="Green" title="E44">A50</font></i></a>, <a class="txt" href="euler_1.html#E12:28_7"><i><font color="Green" title="E45">A51</font></i></a>, <a class="ref" href="euler_1.html#T15" target="_self" title="EULER_1:th.15">Th15</a></i>;<br/>
<a NAME="E15:28_7"/><b>then </b><i><font color="Green" title="E46">A52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">d</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E17:28_7"/><i><font color="Green" title="E47">A53</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c17">d</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E18:28_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E4:28"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="nat_d.html#D2" title="NAT_D:def.2">NAT_D: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" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E15:28_7"><i><font color="Green" title="E46">A52</font></i></a>, <a class="txt" href="euler_1.html#E17:28_7"><i><font color="Green" title="E47">A53</font></i></a>, <a class="ref" href="euler_1.html#T9" target="_self" title="EULER_1:th.9">Th9</a></i>;<br/>
<a NAME="E20:28_7"/><b>then </b>
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c16">t</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E1:28_7"><i><font color="Green" title="E40">A46</font></i></a>, <a class="txt" href="euler_1.html#E5:28_7"><i><font color="Green" title="E41">A47</font></i></a>, <a class="txt" href="euler_1.html#E7:28_7"><i><font color="Green" title="E42">A48</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">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" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c13">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="euler_1.html#E1:28_7"><i><font color="Green" title="E40">A46</font></i></a>, <a class="txt" href="euler_1.html#E7:28_7"><i><font color="Green" title="E42">A48</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:28"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">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" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">A</font> = <font color="Maroon" title="c9">A</font>, <font color="Maroon" title="c13">B</font> = <font color="Maroon" title="c10">B</font>, <font color="Maroon" title="c14">C</font> = <font color="Maroon" title="c11">C</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="euler_1.html#E7:28"><i><font color="Green" title="E33">A7</font></i></a>, <a class="txt" href="euler_1.html#E9:28"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="euler_1.html#E11:28"><i><font color="Green" title="E35">A13</font></i></a>, <a class="txt" href="euler_1.html#E15:28"><i><font color="Green" title="E37">A41</font></i></a>, <a class="txt" href="euler_1.html#E16:28"><i><font color="Green" title="E38">A43</font></i></a>, <a class="txt" href="euler_1.html#E17:28"><i><font color="Green" title="E39">A45</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c12">A</font>,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c13">B</font>) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> $1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> $2</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>;<br/>
<a NAME="E21:28"/><i><font color="Green" title="E40">A54</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c12">A</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c13">B</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">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" title="c15">y</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c12">A</font>;<br/><b>let </b><font color="Maroon" title="c16">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c13">B</font>;<br/>



<a NAME="E1:28_8"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c17">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:28_8"/><i><font color="Green" title="E41">A55</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font>
 <b>and </b><br/><a NAME="E4:28_8"/><i><font color="Green" title="E42">A56</font></i>: 
<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c17">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E5:28_8"/><i><font color="Green" title="E43">A57</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>and </b><br/><a NAME="E6:28_8"/><i><font color="Green" title="E44">A58</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 ;<br/>
<a NAME="E7:28_8"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">B</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c18">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:28_8"/><i><font color="Green" title="E45">A59</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font>
 <b>and </b><br/><a NAME="E10:28_8"/><i><font color="Green" title="E46">A60</font></i>: 
<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c18">j</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E11:28_8"/><i><font color="Green" title="E47">A61</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>and </b><br/><a NAME="E12:28_8"/><i><font color="Green" title="E48">A62</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 ;<br/>
<a NAME="E13:28_8"/><i><font color="Green" title="E49">A63</font></i>: 
( <font color="Maroon" title="c17">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c18">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="euler_1.html#E5:28_8"><i><font color="Green" title="E43">A57</font></i></a>, <a class="txt" href="euler_1.html#E11:28_8"><i><font color="Green" title="E47">A61</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">ii</font> = <font color="Maroon" title="c17">i</font>, <font color="Maroon" title="c20">jj</font> = <font color="Maroon" title="c18">j</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E15:28_8"/>
( <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">j</font> )
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E13:28_8"><i><font color="Green" title="E49">A63</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/>
<a NAME="E16:28_8"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T7" title="NAT_1:th.7">NAT_1:7</a></i>;<br/>
<a NAME="E17:28_8"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
;<br/>
<a NAME="E18:28_8"/><b>then </b><i><font color="Green" title="E50">A64</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">ii</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">jj</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">j</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E6:28"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="euler_1.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E20:28_8"/><i><font color="Green" title="E51">A65</font></i>: 
<font color="Maroon" title="c21">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">ii</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">jj</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 ;<br/>
<a NAME="E21:28_8"/>
<font color="Maroon" title="c4">n</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E4:28_8"><i><font color="Green" title="E42">A56</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E22:28_8"/><b>then </b>
<font color="Maroon" title="c17">i</font>,<font color="Maroon" title="c4">n</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E23:28_8"/><b>then </b>
<font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c4">n</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="euler_1.html#E13:28_8"><i><font color="Green" title="E49">A63</font></i></a>, <a class="ref" href="euler_1.html#T15" target="_self" title="EULER_1:th.15">Th15</a></i>;<br/>
<a NAME="E24:28_8"/><b>then </b><i><font color="Green" title="E52">A66</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E25:28_8"/>
<font color="Maroon" title="c3">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E10:28_8"><i><font color="Green" title="E46">A60</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E26:28_8"/><b>then </b><i><font color="Green" title="E53">A67</font></i>: 
<font color="Maroon" title="c18">j</font>,<font color="Maroon" title="c3">m</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E27:28_8"/>
<font color="Maroon" title="c3">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E28:28_8"/><b>then </b>
<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">m</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E29:28_8"/><b>then </b>
<font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">m</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E13:28_8"><i><font color="Green" title="E49">A63</font></i></a>, <a class="txt" href="euler_1.html#E26:28_8"><i><font color="Green" title="E53">A67</font></i></a>, <a class="ref" href="euler_1.html#T15" target="_self" title="EULER_1:th.15">Th15</a></i>;<br/>
<a NAME="E30:28_8"/><b>then </b><i><font color="Green" title="E54">A68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">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" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">j</font></span>)</span></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E24:28_8"><i><font color="Green" title="E52">A66</font></i></a>, <a class="ref" href="euler_1.html#T9" target="_self" title="EULER_1:th.9">Th9</a></i>;<br/>
<a NAME="E32:28_8"/><b>then </b><i><font color="Green" title="E55">A69</font></i>: 
<font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">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" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">j</font></span>)</span></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E30:28_8"><i><font color="Green" title="E54">A68</font></i></a>, <a class="ref" href="euler_1.html#T9" target="_self" title="EULER_1:th.9">Th9</a></i>;<br/>
<a NAME="E34:28_8"/><b>then </b><i><font color="Green" title="E56">A70</font></i>: 
<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E35:28_8"/>
( <font color="Maroon" title="c17">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c18">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> )
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E13:28_8"><i><font color="Green" title="E49">A63</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/>
<a NAME="E36:28_8"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T7" title="NAT_1:th.7">NAT_1:7</a></i>;<br/>
<a NAME="E37:28_8"/><b>then </b>
<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E32:28_8"><i><font color="Green" title="E55">A69</font></i></a>, <a class="txt" href="euler_1.html#E34:28_8"><i><font color="Green" title="E56">A70</font></i></a>, <a class="ref" href="euler_1.html#T15" target="_self" title="EULER_1:th.15">Th15</a></i>;<br/>
<a NAME="E38:28_8"/><b>then </b><i><font color="Green" title="E57">A71</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E40:28_8"/><i><font color="Green" title="E58">A72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">j</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">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" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">j</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E4:28"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="nat_d.html#D2" title="NAT_D:def.2">NAT_D:def 2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c23">l</font> = <font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">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" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c18">j</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E38:28_8"><i><font color="Green" title="E57">A71</font></i></a>, <a class="txt" href="euler_1.html#E40:28_8"><i><font color="Green" title="E58">A72</font></i></a>, <a class="ref" href="euler_1.html#T9" target="_self" title="EULER_1:th.9">Th9</a></i>;<br/>
<a NAME="E43:28_8"/><b>then </b>
<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c21">q</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E18:28_8"><i><font color="Green" title="E50">A64</font></i></a>, <a class="txt" href="euler_1.html#E20:28_8"><i><font color="Green" title="E51">A65</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">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" title="c15">y</font>,<font color="Maroon" title="c16">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">C</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E3:28_8"><i><font color="Green" title="E41">A55</font></i></a>, <a class="txt" href="euler_1.html#E6:28_8"><i><font color="Green" title="E44">A58</font></i></a>, <a class="txt" href="euler_1.html#E9:28_8"><i><font color="Green" title="E45">A59</font></i></a>, <a class="txt" href="euler_1.html#E12:28_8"><i><font color="Green" title="E48">A62</font></i></a>, <a class="txt" href="euler_1.html#E18:28_8"><i><font color="Green" title="E50">A64</font></i></a>, <a class="txt" href="euler_1.html#E20:28_8"><i><font color="Green" title="E51">A65</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c15">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c13">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c14">C</font><b> such that </b><br/><a NAME="E23:28"/><i><font color="Green" title="E41">A73</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c12">A</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c13">B</font> holds  <font color="Maroon" title="c15">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">x</font>)
 <b>from </b><i><a class="ref" href="funct_7.html#S1" title="FUNCT_7:sch.1">FUNCT_7:sch 1</a>(<a class="txt" href="euler_1.html#E21:28"><i><font color="Green" title="E40">A54</font></i></a>);<br/></i>
<a NAME="E24:28"/><i><font color="Green" title="E42">A74</font></i>: 
<font color="Maroon" title="c15">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">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" title="c16">x</font>, <font color="Maroon" title="c17">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<a NAME="E1:28_9"/><i><font color="Green" title="E43">A75</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c13">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>assume </b><a NAME="E2:28_9"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">xx1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c12">A</font>, <font color="Maroon" title="c19">xx2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c13">B</font><b> such that </b><br/><a NAME="E4:28_9"/><i><font color="Green" title="E44">A76</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">xx1</font>,<font color="Maroon" title="c19">xx2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="euler_1.html#E1:28_9"><i><font color="Green" title="E43">A75</font></i></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E5:28_9"/>
<font color="Maroon" title="c18">xx1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c20">x1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E7:28_9"/><i><font color="Green" title="E45">A77</font></i>: 
<font color="Maroon" title="c18">xx1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x1</font>
 <b>and </b><br/><a NAME="E8:28_9"/>
<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c20">x1</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E9:28_9"/><i><font color="Green" title="E46">A78</font></i>: 
<font color="Maroon" title="c20">x1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>and </b><br/><a NAME="E10:28_9"/><i><font color="Green" title="E47">A79</font></i>: 
<font color="Maroon" title="c20">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 ;<br/>
<a NAME="E11:28_9"/>
<font color="Maroon" title="c19">xx2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">B</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c21">x2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E13:28_9"/><i><font color="Green" title="E48">A80</font></i>: 
<font color="Maroon" title="c19">xx2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x2</font>
 <b>and </b><br/><a NAME="E14:28_9"/>
<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c21">x2</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E15:28_9"/><i><font color="Green" title="E49">A81</font></i>: 
<font color="Maroon" title="c21">x2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>and </b><br/><a NAME="E16:28_9"/><i><font color="Green" title="E50">A82</font></i>: 
<font color="Maroon" title="c21">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 ;<br/>
<b>assume </b><a NAME="E17:28_9"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c22">yy1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c12">A</font>, <font color="Maroon" title="c23">yy2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c13">B</font><b> such that </b><br/><a NAME="E19:28_9"/><i><font color="Green" title="E51">A83</font></i>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">yy1</font>,<font color="Maroon" title="c23">yy2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="euler_1.html#E1:28_9"><i><font color="Green" title="E43">A75</font></i></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E20:28_9"/>
<font color="Maroon" title="c22">yy1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c24">y1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E22:28_9"/><i><font color="Green" title="E52">A84</font></i>: 
<font color="Maroon" title="c22">yy1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">y1</font>
 <b>and </b><br/><a NAME="E23:28_9"/>
<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c24">y1</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E24:28_9"/><i><font color="Green" title="E53">A85</font></i>: 
<font color="Maroon" title="c24">y1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>and </b><br/><a NAME="E25:28_9"/><i><font color="Green" title="E54">A86</font></i>: 
<font color="Maroon" title="c24">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 ;<br/>
<a NAME="E26:28_9"/>
<font color="Maroon" title="c23">yy2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">B</font>
 
;<br/>
<b>then consider </b><font color="Maroon" title="c25">y2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E28:28_9"/><i><font color="Green" title="E55">A87</font></i>: 
<font color="Maroon" title="c23">yy2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">y2</font>
 <b>and </b><br/><a NAME="E29:28_9"/>
<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c25">y2</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E30:28_9"/><i><font color="Green" title="E56">A88</font></i>: 
<font color="Maroon" title="c25">y2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>and </b><br/><a NAME="E31:28_9"/><i><font color="Green" title="E57">A89</font></i>: 
<font color="Maroon" title="c25">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font>
 ;<br/>
<b>assume </b><a NAME="E32:28_9"/><i><font color="Green" title="E58">A90</font></i>: 
<font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font>
 ;<br/>

<i><font color="Green" title="E59">A91</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">x1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c21">x2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> = 
<font color="Maroon" title="c15">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c18">xx1</font>,<font color="Maroon" title="c19">xx2</font>
<b>by </b><i><a class="txt" href="euler_1.html#E23:28"><i><font color="Green" title="E41">A73</font></i></a>, <a class="txt" href="euler_1.html#E7:28_9"><i><font color="Green" title="E45">A77</font></i></a>, <a class="txt" href="euler_1.html#E13:28_9"><i><font color="Green" title="E48">A80</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c15">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">yy1</font>,<font color="Maroon" title="c23">yy2</font>
<b>by </b><i><a class="txt" href="euler_1.html#E32:28_9"><i><font color="Green" title="E58">A90</font></i></a>, <a class="txt" href="euler_1.html#E19:28_9"><i><font color="Green" title="E51">A83</font></i></a>, <a class="txt" href="euler_1.html#E4:28_9"><i><font color="Green" title="E44">A76</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">y1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">y2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
<b>by </b><i><a class="txt" href="euler_1.html#E23:28"><i><font color="Green" title="E41">A73</font></i></a>, <a class="txt" href="euler_1.html#E22:28_9"><i><font color="Green" title="E52">A84</font></i></a>, <a class="txt" href="euler_1.html#E28:28_9"><i><font color="Green" title="E55">A87</font></i></a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c26">y1</font> = <font color="Maroon" title="c24">y1</font>, <font color="Maroon" title="c27">y2</font> = <font color="Maroon" title="c25">y2</font>, <font color="Maroon" title="c28">x1</font> = <font color="Maroon" title="c20">x1</font>, <font color="Maroon" title="c29">x2</font> = <font color="Maroon" title="c21">x2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<b>set </b><font color="Maroon" title="c30">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">x1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>;<br/>
<a NAME="E35:28_9"/><i><font color="Green" title="E60">A92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">y1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">y2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">x1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c26">y1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">y2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E6:28"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="euler_1.html#E33:28_9"><i><font color="Green" title="E59">A91</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/>
<a NAME="E36:28_9"/><i><font color="Green" title="E61">A93</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">x1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c26">y1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">y2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">y1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">y2</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E37:28_9"/><i><font color="Green" title="E62">A94</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" title="c28">x1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c28">x1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">x1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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" title="c26">y1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c27">y2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">x1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c26">y1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">y2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span>
 
;<br/>
<b>set </b><font color="Maroon" title="c31">w</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">x1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c26">y1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c27">y2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>;<br/>
<a NAME="E38:28_9"/>
<font color="Maroon" title="c3">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E39:28_9"/><b>then </b><i><font color="Green" title="E63">A95</font></i>: 
<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c3">m</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c32">p</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E41:28_9"/><i><font color="Green" title="E64">A96</font></i>: 
<font color="Maroon" title="c32">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 ;<br/>
<a NAME="E42:28_9"/><i><font color="Green" title="E65">A97</font></i>: 
<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c32">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E39:28_9"><i><font color="Green" title="E63">A95</font></i></a>, <a class="txt" href="euler_1.html#E41:28_9"><i><font color="Green" title="E64">A96</font></i></a>, <a class="txt" href="euler_1.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<a NAME="E43:28_9"/><i><font color="Green" title="E66">A98</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">y1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">y2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">x1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c26">y1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c27">y2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E6:28"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="euler_1.html#E35:28_9"><i><font color="Green" title="E60">A92</font></i></a>, <a class="txt" href="euler_1.html#E36:28_9"><i><font color="Green" title="E61">A93</font></i></a>, <a class="txt" href="euler_1.html#E37:28_9"><i><font color="Green" title="E62">A94</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/>
<a NAME="E44:28_9"/><b>then </b>
<font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">y2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">x1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c26">y1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c27">y2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c28">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">y1</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E45:28_9"/><b>then </b>
<font color="Maroon" title="c32">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">y2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E41:28_9"><i><font color="Green" title="E64">A96</font></i></a>, <a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E46:28_9"/><b>then </b><i><font color="Green" title="E67">A99</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c29">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">y2</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E41:28_9"><i><font color="Green" title="E64">A96</font></i></a>, <a class="txt" href="euler_1.html#E42:28_9"><i><font color="Green" title="E65">A97</font></i></a>, <a class="ref" href="int_2.html#T40" title="INT_2:th.40">INT_2:40</a></i>;<br/>
<a NAME="E47:28_9"/>
<font color="Maroon" title="c29">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">y2</font> or <font color="Maroon" title="c29">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E68">A100</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">y2</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c33">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:28_9_2_1_1"/><i><font color="Green" title="E69">A101</font></i>: 
<font color="Maroon" title="c33">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">y2</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c34">k</font> = <font color="Maroon" title="c33">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="euler_1.html#E1:28_9_2_1_1"><i><font color="Green" title="E68">A100</font></i></a>, <a class="txt" href="euler_1.html#E3:28_9_2_1_1"><i><font color="Green" title="E69">A101</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E5:28_9_2_1_1"/><i><font color="Green" title="E70">A102</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c34">k</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E46:28_9"><i><font color="Green" title="E67">A99</font></i></a>, <a class="txt" href="euler_1.html#E3:28_9_2_1_1"><i><font color="Green" title="E69">A101</font></i></a>, <a class="txt" href="euler_1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a></i>;<br/><a NAME="E6:28_9_2_1_1"/>
<font color="Maroon" title="c34">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c34">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:28_9_2_1_1_1"/><b>then </b><i><font color="Green" title="E71">A103</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c34">k</font>
 
;<br/>
<a NAME="E3:28_9_2_1_1_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c27">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E30:28_9"><i><font color="Green" title="E56">A88</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<a NAME="E4:28_9_2_1_1_1"/><b>then </b><i><font color="Green" title="E72">A104</font></i>: 
<font color="Maroon" title="c29">x2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c27">y2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E16:28_9"><i><font color="Green" title="E50">A82</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c35">b</font> = <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>;<br/>
<a NAME="E5:28_9_2_1_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E1:28_9_2_1_1"><i><font color="Green" title="E68">A100</font></i></a>, <a class="txt" href="euler_1.html#E4:28_9_2_1_1_1"><i><font color="Green" title="E72">A104</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E6:28_9_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c34">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E3:28_9_2_1_1"><i><font color="Green" title="E69">A101</font></i></a>, <a class="txt" href="euler_1.html#E4:28_9_2_1_1_1"><i><font color="Green" title="E72">A104</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E7:28_9_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_1.html#E5:28_9_2_1_1"><i><font color="Green" title="E70">A102</font></i></a>, <a class="txt" href="euler_1.html#E2:28_9_2_1_1_1"><i><font color="Green" title="E71">A103</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:28_9_2_1_1"/>
<font color="Maroon" title="c29">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="euler_1.html#E3:28_9_2_1_1"><i><font color="Green" title="E69">A101</font></i></a></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" title="E68">A105</font></i>: 
<font color="Maroon" title="c29">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c33">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:28_9_2_1_2"/><i><font color="Green" title="E69">A106</font></i>: 
<font color="Maroon" title="c33">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">y2</font></span>)</span>
 ;<br/><a NAME="E4:28_9_2_1_2"/><i><font color="Green" title="E70">A107</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c33">k</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E46:28_9"><i><font color="Green" title="E67">A99</font></i></a>, <a class="txt" href="euler_1.html#E3:28_9_2_1_2"><i><font color="Green" title="E69">A106</font></i></a>, <a class="ref" href="int_2.html#T14" title="INT_2:th.14">INT_2:14</a></i>;<br/><a NAME="E5:28_9_2_1_2"/><i><font color="Green" title="E71">A108</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c33">k</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E1:28_9_2_1_2"><i><font color="Green" title="E68">A105</font></i></a>, <a class="txt" href="euler_1.html#E3:28_9_2_1_2"><i><font color="Green" title="E69">A106</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c34">k</font> = <font color="Maroon" title="c33">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E7:28_9_2_1_2"/><i><font color="Green" title="E72">A109</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c34">k</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E4:28_9_2_1_2"><i><font color="Green" title="E70">A107</font></i></a>, <a class="txt" href="euler_1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a></i>;<br/><a NAME="E8:28_9_2_1_2"/>
<font color="Maroon" title="c34">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c34">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:28_9_2_1_2_1"/><b>then </b><i><font color="Green" title="E73">A110</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c34">k</font>
 
;<br/>
<a NAME="E3:28_9_2_1_2_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c29">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E15:28_9"><i><font color="Green" title="E49">A81</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<a NAME="E4:28_9_2_1_2_1"/><b>then </b><i><font color="Green" title="E74">A111</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c29">x2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c27">y2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E31:28_9"><i><font color="Green" title="E57">A89</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c35">b</font> = <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>;<br/>
<a NAME="E5:28_9_2_1_2_1"/>
<font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E3:28_9_2_1_2"><i><font color="Green" title="E69">A106</font></i></a>, <a class="txt" href="euler_1.html#E5:28_9_2_1_2"><i><font color="Green" title="E71">A108</font></i></a>, <a class="txt" href="euler_1.html#E4:28_9_2_1_2_1"><i><font color="Green" title="E74">A111</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E6:28_9_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c34">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E3:28_9_2_1_2"><i><font color="Green" title="E69">A106</font></i></a>, <a class="txt" href="euler_1.html#E4:28_9_2_1_2_1"><i><font color="Green" title="E74">A111</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E7:28_9_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_1.html#E7:28_9_2_1_2"><i><font color="Green" title="E72">A109</font></i></a>, <a class="txt" href="euler_1.html#E2:28_9_2_1_2_1"><i><font color="Green" title="E73">A110</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</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" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">y2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="euler_1.html#E3:28_9_2_1_2"><i><font color="Green" title="E69">A106</font></i></a></i>;<br/><b>hence </b><a NAME="E10:28_9_2_1_2"/>
<font color="Maroon" title="c29">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="E68">A112</font></i>: 
<font color="Maroon" title="c29">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">y2</font>
 
;<br/>
<b>consider </b><font color="Maroon" title="c33">p</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E50:28_9"/><i><font color="Green" title="E69">A113</font></i>: 
<font color="Maroon" title="c33">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font>
 ;<br/>
<a NAME="E51:28_9"/><i><font color="Green" title="E70">A114</font></i>: 
<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c33">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="euler_1.html#E50:28_9"><i><font color="Green" title="E69">A113</font></i></a>, <a class="txt" href="euler_1.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<a NAME="E52:28_9"/>
<font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">x1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c26">y1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c27">y2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">x2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c27">y2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E43:28_9"><i><font color="Green" title="E66">A98</font></i></a></i>;<br/>
<a NAME="E53:28_9"/><b>then </b>
<font color="Maroon" title="c33">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">y1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E50:28_9"><i><font color="Green" title="E69">A113</font></i></a>, <a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E54:28_9"/><b>then </b><i><font color="Green" title="E71">A115</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c28">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">y1</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E50:28_9"><i><font color="Green" title="E69">A113</font></i></a>, <a class="txt" href="euler_1.html#E51:28_9"><i><font color="Green" title="E70">A114</font></i></a>, <a class="ref" href="int_2.html#T40" title="INT_2:th.40">INT_2:40</a></i>;<br/>
<a NAME="E55:28_9"/>
<font color="Maroon" title="c28">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">y1</font> or <font color="Maroon" title="c28">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E72">A116</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c28">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">y1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c34">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:28_9_3_1_1"/><i><font color="Green" title="E73">A117</font></i>: 
<font color="Maroon" title="c34">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">y1</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c35">k</font> = <font color="Maroon" title="c34">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="euler_1.html#E1:28_9_3_1_1"><i><font color="Green" title="E72">A116</font></i></a>, <a class="txt" href="euler_1.html#E3:28_9_3_1_1"><i><font color="Green" title="E73">A117</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E5:28_9_3_1_1"/><i><font color="Green" title="E74">A118</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c35">k</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E54:28_9"><i><font color="Green" title="E71">A115</font></i></a>, <a class="txt" href="euler_1.html#E3:28_9_3_1_1"><i><font color="Green" title="E73">A117</font></i></a>, <a class="txt" href="euler_1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a></i>;<br/><a NAME="E6:28_9_3_1_1"/>
<font color="Maroon" title="c35">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c35">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:28_9_3_1_1_1"/><b>then </b><i><font color="Green" title="E75">A119</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">k</font>
 
;<br/>
<a NAME="E3:28_9_3_1_1_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c26">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E24:28_9"><i><font color="Green" title="E53">A85</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<a NAME="E4:28_9_3_1_1_1"/><b>then </b><i><font color="Green" title="E76">A120</font></i>: 
<font color="Maroon" title="c28">x1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c26">y1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E10:28_9"><i><font color="Green" title="E47">A79</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c36">b</font> = <font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>;<br/>
<a NAME="E5:28_9_3_1_1_1"/>
<font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E1:28_9_3_1_1"><i><font color="Green" title="E72">A116</font></i></a>, <a class="txt" href="euler_1.html#E4:28_9_3_1_1_1"><i><font color="Green" title="E76">A120</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E6:28_9_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c35">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E3:28_9_3_1_1"><i><font color="Green" title="E73">A117</font></i></a>, <a class="txt" href="euler_1.html#E4:28_9_3_1_1_1"><i><font color="Green" title="E76">A120</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</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#E5:28_9_3_1_1"><i><font color="Green" title="E74">A118</font></i></a>, <a class="txt" href="euler_1.html#E2:28_9_3_1_1_1"><i><font color="Green" title="E75">A119</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:28_9_3_1_1"/>
<font color="Maroon" title="c28">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="euler_1.html#E3:28_9_3_1_1"><i><font color="Green" title="E73">A117</font></i></a></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" title="E72">A121</font></i>: 
<font color="Maroon" title="c28">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c34">k</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:28_9_3_1_2"/><i><font color="Green" title="E73">A122</font></i>: 
<font color="Maroon" title="c34">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">y1</font></span>)</span>
 ;<br/><a NAME="E4:28_9_3_1_2"/><i><font color="Green" title="E74">A123</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c34">k</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E54:28_9"><i><font color="Green" title="E71">A115</font></i></a>, <a class="txt" href="euler_1.html#E3:28_9_3_1_2"><i><font color="Green" title="E73">A122</font></i></a>, <a class="ref" href="int_2.html#T14" title="INT_2:th.14">INT_2:14</a></i>;<br/><a NAME="E5:28_9_3_1_2"/><i><font color="Green" title="E75">A124</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c34">k</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E1:28_9_3_1_2"><i><font color="Green" title="E72">A121</font></i></a>, <a class="txt" href="euler_1.html#E3:28_9_3_1_2"><i><font color="Green" title="E73">A122</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c35">k</font> = <font color="Maroon" title="c34">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E7:28_9_3_1_2"/><i><font color="Green" title="E76">A125</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c35">k</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E4:28_9_3_1_2"><i><font color="Green" title="E74">A123</font></i></a>, <a class="txt" href="euler_1.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a></i>;<br/><a NAME="E8:28_9_3_1_2"/>
<font color="Maroon" title="c35">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c35">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:28_9_3_1_2_1"/><b>then </b><i><font color="Green" title="E77">A126</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c35">k</font>
 
;<br/>
<a NAME="E3:28_9_3_1_2_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c28">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E9:28_9"><i><font color="Green" title="E46">A78</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<a NAME="E4:28_9_3_1_2_1"/><b>then </b><i><font color="Green" title="E78">A127</font></i>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c28">x1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c26">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E25:28_9"><i><font color="Green" title="E54">A86</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<b>set </b><font color="Maroon" title="c36">b</font> = <font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>;<br/>
<a NAME="E5:28_9_3_1_2_1"/>
<font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="txt" href="euler_1.html#E3:28_9_3_1_2"><i><font color="Green" title="E73">A122</font></i></a>, <a class="txt" href="euler_1.html#E5:28_9_3_1_2"><i><font color="Green" title="E75">A124</font></i></a>, <a class="txt" href="euler_1.html#E4:28_9_3_1_2_1"><i><font color="Green" title="E78">A127</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E6:28_9_3_1_2_1"/><b>then </b>
<font color="Maroon" title="c35">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E3:28_9_3_1_2"><i><font color="Green" title="E73">A122</font></i></a>, <a class="txt" href="euler_1.html#E4:28_9_3_1_2_1"><i><font color="Green" title="E78">A127</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E7:28_9_3_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_1.html#E7:28_9_3_1_2"><i><font color="Green" title="E76">A125</font></i></a>, <a class="txt" href="euler_1.html#E2:28_9_3_1_2_1"><i><font color="Green" title="E77">A126</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</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" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">y1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="euler_1.html#E3:28_9_3_1_2"><i><font color="Green" title="E73">A122</font></i></a></i>;<br/><b>hence </b><a NAME="E10:28_9_3_1_2"/>
<font color="Maroon" title="c28">x1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c26">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E4:28_9"><i><font color="Green" title="E44">A76</font></i></a>, <a class="txt" href="euler_1.html#E7:28_9"><i><font color="Green" title="E45">A77</font></i></a>, <a class="txt" href="euler_1.html#E13:28_9"><i><font color="Green" title="E48">A80</font></i></a>, <a class="txt" href="euler_1.html#E19:28_9"><i><font color="Green" title="E51">A83</font></i></a>, <a class="txt" href="euler_1.html#E22:28_9"><i><font color="Green" title="E52">A84</font></i></a>, <a class="txt" href="euler_1.html#E28:28_9"><i><font color="Green" title="E55">A87</font></i></a>, <a class="txt" href="euler_1.html#E48:28_9"><i><font color="Green" title="E68">A112</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:28"/>
<font color="Maroon" title="c15">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">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" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">C</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c16">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:28_10"/>
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:28_10"/><i><font color="Green" title="E43">A128</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">z</font>
 <b>and </b><br/><a NAME="E5:28_10"/><i><font color="Green" title="E44">A129</font></i>: 
 ex <font color="Olive" title="b1">x0</font>, <font color="Olive" title="b2">y0</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> &amp; <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c17">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Olive" title="b2">y0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Olive" title="b1">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 ;<br/>
<b>consider </b><font color="Maroon" title="c18">x0</font>, <font color="Maroon" title="c19">y0</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E7:28_10"/><i><font color="Green" title="E45">A130</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 <b>and </b><br/><a NAME="E8:28_10"/><i><font color="Green" title="E46">A131</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c17">i</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E9:28_10"/>
( <font color="Maroon" title="c19">y0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c18">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 <b>by </b><i><a class="txt" href="euler_1.html#E5:28_10"><i><font color="Green" title="E44">A129</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">x</font>, <font color="Maroon" title="c21">y</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E11:28_10"/><i><font color="Green" title="E47">A132</font></i>: 
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x0</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">m</font>
 <b>and </b><br/><a NAME="E12:28_10"/><i><font color="Green" title="E48">A133</font></i>: 
<font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y0</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">n</font>
 ;<br/>
<a NAME="E13:28_10"/><i><font color="Green" title="E49">A134</font></i>: 
( <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c21">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> )
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E11:28_10"><i><font color="Green" title="E47">A132</font></i></a>, <a class="txt" href="euler_1.html#E12:28_10"><i><font color="Green" title="E48">A133</font></i></a>, <a class="ref" href="newton.html#T79" title="NEWTON:th.79">NEWTON:79</a></i>;<br/>
<a NAME="E14:28_10"/><i><font color="Green" title="E50">A135</font></i>: 
( <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c21">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 
<b>by </b><i><a class="txt" href="euler_1.html#E11:28_10"><i><font color="Green" title="E47">A132</font></i></a>, <a class="txt" href="euler_1.html#E12:28_10"><i><font color="Green" title="E48">A133</font></i></a>, <a class="ref" href="newton.html#T78" title="NEWTON:th.78">NEWTON:78</a></i>;<br/>
<a NAME="E15:28_10"/><i><font color="Green" title="E51">A136</font></i>: 
<font color="Maroon" title="c21">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="euler_1.html#E12:28_10"><i><font color="Green" title="E48">A133</font></i></a>, <a class="ref" href="newton.html#T78" title="NEWTON:th.78">NEWTON:78</a></i>;<br/>
<a NAME="E16:28_10"/><i><font color="Green" title="E52">A137</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c21">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">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" title="E53">A138</font></i>: 
<font color="Maroon" title="c18">x0</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c22">j</font> = <font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font>;<br/>
<a NAME="E2:28_10_1"/><i><font color="Green" title="E54">A139</font></i>: 
<font color="Maroon" title="c19">y0</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c23">k</font> = <font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font>;<br/>
<a NAME="E3:28_10_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c21">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E11:28_10"><i><font color="Green" title="E47">A132</font></i></a>, <a class="txt" href="euler_1.html#E12:28_10"><i><font color="Green" title="E48">A133</font></i></a>, <a class="txt" href="euler_1.html#E1:28_10_1"><i><font color="Green" title="E53">A138</font></i></a>, <a class="txt" href="euler_1.html#E2:28_10_1"><i><font color="Green" title="E54">A139</font></i></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" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c21">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">x</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="euler_1.html#T13" target="_self" title="EULER_1:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:28_10"/><i><font color="Green" title="E53">A140</font></i>: 
<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c18">x0</font> <a href="int_2.html#R1" title="INT_2:pred.1">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" title="c22">p</font> = <font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font>;<br/>
<a NAME="E1:28_10_2"/><i><font color="Green" title="E54">A141</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#T29" title="INT_2:th.29">INT_2:29</a></i>;<br/>
<a NAME="E2:28_10_2"/>
<font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T31" title="INT_2:th.31">INT_2:31</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c23">i1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E4:28_10_2"/><i><font color="Green" title="E55">A142</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E5:28_10_2"/>
( <font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c23">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="euler_1.html#E4:28_10_2"><i><font color="Green" title="E55">A142</font></i></a></i>;<br/>
<a NAME="E6:28_10_2"/><b>then </b><i><font color="Green" title="E56">A143</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="euler_1.html#E1:28_10_2"><i><font color="Green" title="E54">A141</font></i></a></i>;<br/>
<a NAME="E7:28_10_2"/>
<font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c18">x0</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T32" title="INT_2:th.32">INT_2:32</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c24">i2</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E9:28_10_2"/><i><font color="Green" title="E57">A144</font></i>: 
<font color="Maroon" title="c18">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E10:28_10_2"/><i><font color="Green" title="E58">A145</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E8:28_10"><i><font color="Green" title="E46">A131</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E11:28_10_2"/>
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E12:28_10_2"/><b>then </b><i><font color="Green" title="E59">A146</font></i>: 
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E10:28_10_2"><i><font color="Green" title="E58">A145</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E13:28_10_2"/><i><font color="Green" title="E60">A147</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E4:28"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="euler_1.html#E4:28_10_2"><i><font color="Green" title="E55">A142</font></i></a>, <a class="txt" href="euler_1.html#E9:28_10_2"><i><font color="Green" title="E57">A144</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">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" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span>;<br/>
<a NAME="E14:28_10_2"/><i><font color="Green" title="E61">A148</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" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
 
;<br/>
<b>set </b><font color="Maroon" title="c26">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">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" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/>

<a NAME="E2:28_10_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E6:28_10_2"><i><font color="Green" title="E56">A143</font></i></a>, <a class="ref" href="xreal_1.html#T71" title="XREAL_1:th.71">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#E4:28"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="euler_1.html#E7:28_10"><i><font color="Green" title="E45">A130</font></i></a>, <a class="txt" href="euler_1.html#E4:28_10_2"><i><font color="Green" title="E55">A142</font></i></a>, <a class="txt" href="euler_1.html#E9:28_10_2"><i><font color="Green" title="E57">A144</font></i></a>, <a class="txt" href="euler_1.html#E14:28_10_2"><i><font color="Green" title="E61">A148</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c27">p</font> = <font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font>, <font color="Maroon" title="c28">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a>, <a class="ref" href="int_2.html#T29" title="INT_2:th.29">INT_2:29</a></i>;<br/>
<a NAME="E17:28_10_2"/><i><font color="Green" title="E62">A149</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E4:28_10_2"><i><font color="Green" title="E55">A142</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c29">jj</font> = <font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font>;<br/>
<a NAME="E18:28_10_2"/>
not <font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/>

<a NAME="E2:28_10_2_2"/><b>then </b>
<font color="Maroon" title="c27">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E6:28_10_2"><i><font color="Green" title="E56">A143</font></i></a>, <a class="ref" href="xreal_1.html#T71" title="XREAL_1:th.71">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#E5:28"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="euler_1.html#E4:28_10_2"><i><font color="Green" title="E55">A142</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c30">jj</font> = <font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">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" title="c27">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c30">jj</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c28">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 ) or ( <font color="Maroon" title="c27">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c30">jj</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) )
 </span><b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E17:28_10_2"><i><font color="Green" title="E62">A149</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">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" title="E63">A150</font></i>: 
( <font color="Maroon" title="c27">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c30">jj</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c28">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:28_10_2_3_1_1"/><b>then </b><i><font color="Green" title="E64">A151</font></i>: 
<font color="Maroon" title="c27">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><a NAME="E3:28_10_2_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c27">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c30">jj</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c28">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="euler_1.html#E7:28_10"><i><font color="Green" title="E45">A130</font></i></a>, <a class="txt" href="euler_1.html#E4:28_10_2"><i><font color="Green" title="E55">A142</font></i></a>, <a class="txt" href="euler_1.html#E12:28_10_2"><i><font color="Green" title="E59">A146</font></i></a>, <a class="txt" href="euler_1.html#E13:28_10_2"><i><font color="Green" title="E60">A147</font></i></a>, <a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></i>;<br/><a NAME="E4:28_10_2_3_1_1"/><b>then </b><i><font color="Green" title="E65">A152</font></i>: 
<font color="Maroon" title="c27">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">jj</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c28">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="euler_1.html#E1:28_10_2_3_1_1"><i><font color="Green" title="E63">A150</font></i></a>, <a class="txt" href="euler_1.html#E2:28_10_2_3_1_1"><i><font color="Green" title="E64">A151</font></i></a>, <a class="ref" href="euler_1.html#T16" target="_self" title="EULER_1:th.16">Th16</a></i>;<br/><a NAME="E5:28_10_2_3_1_1"/>
<font color="Maroon" title="c30">jj</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c28">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="int_2.html#T29" title="INT_2:th.29">INT_2:29</a></i>;<br/><b>hence </b><a NAME="E6:28_10_2_3_1_1"/>
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="euler_1.html#E4:28_10_2_3_1_1"><i><font color="Green" title="E65">A152</font></i></a>, <a class="ref" href="nat_1.html#T15" title="NAT_1:th.15">NAT_1:15</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" title="c27">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c30">jj</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c27">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c30">jj</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="euler_1.html#E7:28_10"><i><font color="Green" title="E45">A130</font></i></a>, <a class="txt" href="euler_1.html#E8:28_10"><i><font color="Green" title="E46">A131</font></i></a>, <a class="txt" href="euler_1.html#E4:28_10_2"><i><font color="Green" title="E55">A142</font></i></a>, <a class="txt" href="euler_1.html#E13:28_10_2"><i><font color="Green" title="E60">A147</font></i></a>, <a class="txt" href="euler_1.html#E14:28_10_2"><i><font color="Green" title="E61">A148</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/><a NAME="E3:28_10_2_3_1_2"/><b>then </b>
<font color="Maroon" title="c27">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c30">jj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T65" title="NEWTON:th.65">NEWTON:65</a></i>;<br/><b>hence </b><a NAME="E4:28_10_2_3_1_2"/>
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T15" title="NAT_1:th.15">NAT_1:15</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" title="c3">m</font>,<font color="Maroon" title="c18">x0</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:28_10"/><i><font color="Green" title="E54">A153</font></i>: 
<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c19">y0</font> <a href="int_2.html#R1" title="INT_2:pred.1">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" title="c22">p</font> = <font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font>;<br/>
<a NAME="E1:28_10_3"/><i><font color="Green" title="E55">A154</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#T29" title="INT_2:th.29">INT_2:29</a></i>;<br/>
<a NAME="E2:28_10_3"/>
<font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T31" title="INT_2:th.31">INT_2:31</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c23">i1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E4:28_10_3"/><i><font color="Green" title="E56">A155</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E5:28_10_3"/>
( <font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c23">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="euler_1.html#E1:28"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="euler_1.html#E4:28_10_3"><i><font color="Green" title="E56">A155</font></i></a></i>;<br/>
<a NAME="E6:28_10_3"/><b>then </b><i><font color="Green" title="E57">A156</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="euler_1.html#E1:28_10_3"><i><font color="Green" title="E55">A154</font></i></a></i>;<br/>
<a NAME="E7:28_10_3"/>
<font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c19">y0</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T32" title="INT_2:th.32">INT_2:32</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c24">i2</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E9:28_10_3"/><i><font color="Green" title="E58">A157</font></i>: 
<font color="Maroon" title="c19">y0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font>
 <b>by </b><i><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></i>;<br/>
<a NAME="E10:28_10_3"/><i><font color="Green" title="E59">A158</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E8:28_10"><i><font color="Green" title="E46">A131</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E11:28_10_3"/>
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E12:28_10_3"/><b>then </b><i><font color="Green" title="E60">A159</font></i>: 
<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E10:28_10_3"><i><font color="Green" title="E59">A158</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E13:28_10_3"/><i><font color="Green" title="E61">A160</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E4:28"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="euler_1.html#E4:28_10_3"><i><font color="Green" title="E56">A155</font></i></a>, <a class="txt" href="euler_1.html#E9:28_10_3"><i><font color="Green" title="E58">A157</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span></span>)</span>;<br/>
<a NAME="E14:28_10_3"/><i><font color="Green" title="E62">A161</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span>
 
;<br/>
<b>set </b><font color="Maroon" title="c26">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">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" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&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" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/>

<a NAME="E2:28_10_3_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E6:28_10_3"><i><font color="Green" title="E57">A156</font></i></a>, <a class="ref" href="xreal_1.html#T71" title="XREAL_1:th.71">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#E4:28"><i><font color="Green" title="E30">A4</font></i></a>, <a class="txt" href="euler_1.html#E7:28_10"><i><font color="Green" title="E45">A130</font></i></a>, <a class="txt" href="euler_1.html#E4:28_10_3"><i><font color="Green" title="E56">A155</font></i></a>, <a class="txt" href="euler_1.html#E9:28_10_3"><i><font color="Green" title="E58">A157</font></i></a>, <a class="txt" href="euler_1.html#E14:28_10_3"><i><font color="Green" title="E62">A161</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c27">p</font> = <font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font>, <font color="Maroon" title="c28">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c24">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c23">i1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a>, <a class="ref" href="int_2.html#T29" title="INT_2:th.29">INT_2:29</a></i>;<br/>
<a NAME="E17:28_10_3"/><i><font color="Green" title="E63">A162</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E4:28_10_3"><i><font color="Green" title="E56">A155</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c29">jj</font> = <font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font>;<br/>
<a NAME="E18:28_10_3"/>
not <font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/>

<a NAME="E2:28_10_3_2"/><b>then </b>
<font color="Maroon" title="c27">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E6:28_10_3"><i><font color="Green" title="E57">A156</font></i></a>, <a class="ref" href="xreal_1.html#T71" title="XREAL_1:th.71">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#E5:28"><i><font color="Green" title="E31">A5</font></i></a>, <a class="txt" href="euler_1.html#E4:28_10_3"><i><font color="Green" title="E56">A155</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c30">jj</font> = <font color="Maroon" title="c23">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">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" title="c27">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c30">jj</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c28">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 ) or ( <font color="Maroon" title="c27">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c30">jj</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) )
 </span><b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E17:28_10_3"><i><font color="Green" title="E63">A162</font></i></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">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" title="E64">A163</font></i>: 
( <font color="Maroon" title="c27">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c30">jj</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c28">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:28_10_3_3_1_1"/><b>then </b><i><font color="Green" title="E65">A164</font></i>: 
<font color="Maroon" title="c27">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><a NAME="E3:28_10_3_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c27">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c30">jj</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c28">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="euler_1.html#E7:28_10"><i><font color="Green" title="E45">A130</font></i></a>, <a class="txt" href="euler_1.html#E4:28_10_3"><i><font color="Green" title="E56">A155</font></i></a>, <a class="txt" href="euler_1.html#E12:28_10_3"><i><font color="Green" title="E60">A159</font></i></a>, <a class="txt" href="euler_1.html#E13:28_10_3"><i><font color="Green" title="E61">A160</font></i></a>, <a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></i>;<br/><a NAME="E4:28_10_3_3_1_1"/><b>then </b><i><font color="Green" title="E66">A165</font></i>: 
<font color="Maroon" title="c27">p</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">jj</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c28">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="euler_1.html#E1:28_10_3_3_1_1"><i><font color="Green" title="E64">A163</font></i></a>, <a class="txt" href="euler_1.html#E2:28_10_3_3_1_1"><i><font color="Green" title="E65">A164</font></i></a>, <a class="ref" href="euler_1.html#T16" target="_self" title="EULER_1:th.16">Th16</a></i>;<br/><a NAME="E5:28_10_3_3_1_1"/>
<font color="Maroon" title="c30">jj</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c28">j</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="int_2.html#T29" title="INT_2:th.29">INT_2:29</a></i>;<br/><b>hence </b><a NAME="E6:28_10_3_3_1_1"/>
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="euler_1.html#E4:28_10_3_3_1_1"><i><font color="Green" title="E66">A165</font></i></a>, <a class="ref" href="nat_1.html#T15" title="NAT_1:th.15">NAT_1:15</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" title="c27">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c30">jj</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 &amp; <font color="Maroon" title="c28">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c27">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c30">jj</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="euler_1.html#E7:28_10"><i><font color="Green" title="E45">A130</font></i></a>, <a class="txt" href="euler_1.html#E8:28_10"><i><font color="Green" title="E46">A131</font></i></a>, <a class="txt" href="euler_1.html#E4:28_10_3"><i><font color="Green" title="E56">A155</font></i></a>, <a class="txt" href="euler_1.html#E13:28_10_3"><i><font color="Green" title="E61">A160</font></i></a>, <a class="txt" href="euler_1.html#E14:28_10_3"><i><font color="Green" title="E62">A161</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/><a NAME="E3:28_10_3_3_1_2"/><b>then </b>
<font color="Maroon" title="c27">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c30">jj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T65" title="NEWTON:th.65">NEWTON:65</a></i>;<br/><b>hence </b><a NAME="E4:28_10_3_3_1_2"/>
<font color="Maroon" title="c27">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T15" title="NAT_1:th.15">NAT_1:15</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" title="c4">n</font>,<font color="Maroon" title="c19">y0</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c22">x</font> = <font color="Maroon" title="c20">x</font>, <font color="Maroon" title="c23">y</font> = <font color="Maroon" title="c21">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="euler_1.html#E14:28_10"><i><font color="Green" title="E50">A135</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E20:28_10"/><i><font color="Green" title="E55">A166</font></i>: 
<font color="Maroon" title="c22">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&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" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<a NAME="E2:28_10_4"/><b>then </b><i><font color="Green" title="E56">A167</font></i>: 
<font color="Maroon" title="c18">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E11:28_10"><i><font color="Green" title="E47">A132</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c24">j</font> = <font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font>;<br/>
<i><font color="Green" title="E57">A168</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euler_1.html#E2:28_10_4"><i><font color="Green" title="E56">A167</font></i></a>, <a class="txt" href="euler_1.html#E4:28"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<b>set </b><font color="Maroon" title="c25">jj</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c26">g</font> = <font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="euler_1.html#E7:28_10"><i><font color="Green" title="E45">A130</font></i></a>, <a class="txt" href="euler_1.html#E3:28_10_4"><i><font color="Green" title="E57">A168</font></i></a></i>;<br/>
<a NAME="E5:28_10_4"/><i><font color="Green" title="E58">A169</font></i>: 
( <font color="Maroon" title="c26">g</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c26">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
;<br/>
<a NAME="E6:28_10_4"/><i><font color="Green" title="E59">A170</font></i>: 
not <span class="p1">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/>

<a NAME="E2:28_10_4_2"/><b>then </b>
<font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="xreal_1.html#T71" title="XREAL_1:th.71">XREAL_1:71</a></i>;<br/>
<b>hence </b><a NAME="E3:28_10_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_1.html#E5:28_10_4"><i><font color="Green" title="E58">A169</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:28_10_4"/><i><font color="Green" title="E60">A171</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E8:28_10"><i><font color="Green" title="E46">A131</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E8:28_10_4"/>
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">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" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E7:28_10_4"><i><font color="Green" title="E60">A171</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E10:28_10_4"/><b>then </b><i><font color="Green" title="E61">A172</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E7:28_10"><i><font color="Green" title="E45">A130</font></i></a>, <a class="txt" href="euler_1.html#E3:28_10_4"><i><font color="Green" title="E57">A168</font></i></a>, <a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">jj</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="euler_1.html#E6:28_10_4"><i><font color="Green" title="E59">A170</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E12:28_10_4"/>
<font color="Maroon" title="c27">jj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c27">jj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<a NAME="E2:28_10_4_3"/><b>then </b><i><font color="Green" title="E62">A173</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c27">jj</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E10:28_10_4"><i><font color="Green" title="E61">A172</font></i></a>, <a class="ref" href="euler_1.html#T16" target="_self" title="EULER_1:th.16">Th16</a></i>;<br/>
<a NAME="E3:28_10_4_3"/>
<font color="Maroon" title="c4">n</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c27">jj</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#T29" title="INT_2:th.29">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#E1:28"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="euler_1.html#E2:28_10_4_3"><i><font color="Green" title="E62">A173</font></i></a>, <a class="ref" href="nat_1.html#T15" title="NAT_1:th.15">NAT_1:15</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" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E7:28_10"><i><font color="Green" title="E45">A130</font></i></a>, <a class="txt" href="euler_1.html#E8:28_10"><i><font color="Green" title="E46">A131</font></i></a>, <a class="txt" href="euler_1.html#E3:28_10_4"><i><font color="Green" title="E57">A168</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">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#E3:28"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="newton.html#T65" title="NEWTON:th.65">NEWTON:65</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:28_10"/><b>then </b><i><font color="Green" title="E56">A174</font></i>: 
<font color="Maroon" title="c22">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E14:28_10"><i><font color="Green" title="E50">A135</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/>
<a NAME="E22:28_10"/><i><font color="Green" title="E57">A175</font></i>: 
<font color="Maroon" title="c23">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&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" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<a NAME="E2:28_10_5"/><b>then </b><i><font color="Green" title="E58">A176</font></i>: 
<font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E12:28_10"><i><font color="Green" title="E48">A133</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c24">j</font> = <font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font>;<br/>
<i><font color="Green" title="E59">A177</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">y0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euler_1.html#E2:28_10_5"><i><font color="Green" title="E58">A176</font></i></a>, <a class="txt" href="euler_1.html#E4:28"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<b>set </b><font color="Maroon" title="c25">jj</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c26">g</font> = <font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="euler_1.html#E7:28_10"><i><font color="Green" title="E45">A130</font></i></a>, <a class="txt" href="euler_1.html#E3:28_10_5"><i><font color="Green" title="E59">A177</font></i></a></i>;<br/>
<a NAME="E5:28_10_5"/><i><font color="Green" title="E60">A178</font></i>: 
( <font color="Maroon" title="c26">g</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c26">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
;<br/>
<a NAME="E6:28_10_5"/><i><font color="Green" title="E61">A179</font></i>: 
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0
 ;<br/>

<a NAME="E2:28_10_5_2"/><b>then </b>
<font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="xreal_1.html#T71" title="XREAL_1:th.71">XREAL_1:71</a></i>;<br/>
<b>hence </b><a NAME="E3:28_10_5_2"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_1.html#E5:28_10_5"><i><font color="Green" title="E60">A178</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:28_10_5"/><i><font color="Green" title="E62">A180</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E8:28_10"><i><font color="Green" title="E46">A131</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<a NAME="E8:28_10_5"/>
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">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" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E7:28_10_5"><i><font color="Green" title="E62">A180</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E10:28_10_5"/><b>then </b><i><font color="Green" title="E63">A181</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E7:28_10"><i><font color="Green" title="E45">A130</font></i></a>, <a class="txt" href="euler_1.html#E3:28_10_5"><i><font color="Green" title="E59">A177</font></i></a>, <a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c27">jj</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c18">x0</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="euler_1.html#E6:28_10_5"><i><font color="Green" title="E61">A179</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E12:28_10_5"/>
<font color="Maroon" title="c27">jj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c27">jj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<a NAME="E2:28_10_5_3"/><b>then </b><i><font color="Green" title="E64">A182</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c27">jj</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E10:28_10_5"><i><font color="Green" title="E63">A181</font></i></a>, <a class="ref" href="euler_1.html#T16" target="_self" title="EULER_1:th.16">Th16</a></i>;<br/>
<a NAME="E3:28_10_5_3"/>
<font color="Maroon" title="c3">m</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c27">jj</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#T29" title="INT_2:th.29">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#E1:28"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="euler_1.html#E2:28_10_5_3"><i><font color="Green" title="E64">A182</font></i></a>, <a class="ref" href="nat_1.html#T15" title="NAT_1:th.15">NAT_1:15</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" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E7:28_10"><i><font color="Green" title="E45">A130</font></i></a>, <a class="txt" href="euler_1.html#E8:28_10"><i><font color="Green" title="E46">A131</font></i></a>, <a class="txt" href="euler_1.html#E3:28_10_5"><i><font color="Green" title="E59">A177</font></i></a>, <a class="ref" href="int_2.html#D6" title="INT_2:def.6">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#E3:28"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="newton.html#T65" title="NEWTON:th.65">NEWTON:65</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:28_10"/><b>then </b><i><font color="Green" title="E58">A183</font></i>: 
<font color="Maroon" title="c23">y</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E15:28_10"><i><font color="Green" title="E51">A136</font></i></a>, <a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></i>;<br/>
<a NAME="E24:28_10"/><i><font color="Green" title="E59">A184</font></i>: 
<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c22">x</font> <a href="int_2.html#R2" title="INT_2:pred.2">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" title="E60">A185</font></i>: 
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E11:28_10"><i><font color="Green" title="E47">A132</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>;<br/>
<a NAME="E2:28_10_6"/>
<font color="Maroon" title="c18">x0</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">x0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E1:28_10_6"><i><font color="Green" title="E60">A185</font></i></a></i>;<br/>
<a NAME="E3:28_10_6"/><b>then </b><i><font color="Green" title="E61">A186</font></i>: 
<font color="Maroon" title="c18">x0</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E20:28_10"><i><font color="Green" title="E55">A166</font></i></a>, <a class="ref" href="euler_1.html#T17" target="_self" title="EULER_1:th.17">Th17</a></i>;<br/>
<a NAME="E4:28_10_6"/>
<font color="Maroon" title="c18">x0</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E17:28_10"><i><font color="Green" title="E53">A140</font></i></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></i>;<br/>
<a NAME="E5:28_10_6"/><b>then </b>
<font color="Maroon" title="c3">m</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E3:28_10_6"><i><font color="Green" title="E61">A186</font></i></a>, <a class="txt" href="euler_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:28_10_6"/>
<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c22">x</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:28_10"/><i><font color="Green" title="E60">A187</font></i>: 
<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c23">y</font> <a href="int_2.html#R2" title="INT_2:pred.2">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" title="E61">A188</font></i>: 
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y0</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E12:28_10"><i><font color="Green" title="E48">A133</font></i></a>, <a class="ref" href="int_1.html#D8" title="INT_1:def.8">INT_1:def 8</a></i>;<br/>
<a NAME="E2:28_10_7"/>
<font color="Maroon" title="c19">y0</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">y</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">y0</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E1:28_10_7"><i><font color="Green" title="E61">A188</font></i></a></i>;<br/>
<a NAME="E3:28_10_7"/><b>then </b><i><font color="Green" title="E62">A189</font></i>: 
<font color="Maroon" title="c19">y0</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">y</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E22:28_10"><i><font color="Green" title="E57">A175</font></i></a>, <a class="ref" href="euler_1.html#T17" target="_self" title="EULER_1:th.17">Th17</a></i>;<br/>
<a NAME="E4:28_10_7"/>
<font color="Maroon" title="c19">y0</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E18:28_10"><i><font color="Green" title="E54">A153</font></i></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></i>;<br/>
<a NAME="E5:28_10_7"/><b>then </b>
<font color="Maroon" title="c4">n</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c23">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_1.html#E3:28_10_7"><i><font color="Green" title="E62">A189</font></i></a>, <a class="txt" href="euler_1.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:28_10_7"/>
<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c23">y</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:28_10"/>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">B</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E13:28_10"><i><font color="Green" title="E49">A134</font></i></a>, <a class="txt" href="euler_1.html#E21:28_10"><i><font color="Green" title="E56">A174</font></i></a>, <a class="txt" href="euler_1.html#E24:28_10"><i><font color="Green" title="E59">A184</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c24">x</font> = <font color="Maroon" title="c22">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c13">B</font> ;<br/>
<a NAME="E28:28_10"/>
<font color="Maroon" title="c23">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">A</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E13:28_10"><i><font color="Green" title="E49">A134</font></i></a>, <a class="txt" href="euler_1.html#E23:28_10"><i><font color="Green" title="E58">A183</font></i></a>, <a class="txt" href="euler_1.html#E25:28_10"><i><font color="Green" title="E60">A187</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c25">y</font> = <font color="Maroon" title="c23">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c12">A</font> ;<br/>
<a NAME="E30:28_10"/>
<font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c25">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0
 
<b>by </b><i><a class="txt" href="euler_1.html#E2:28"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="euler_1.html#E15:28_10"><i><font color="Green" title="E51">A136</font></i></a>, <a class="txt" href="euler_1.html#E22:28_10"><i><font color="Green" title="E57">A175</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E31:28_10"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c25">y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c24">x</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E32:28_10"/><b>then </b><i><font color="Green" title="E61">A190</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c25">y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c24">x</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_1.html#E6:28"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="euler_1.html#E7:28_10"><i><font color="Green" title="E45">A130</font></i></a>, <a class="txt" href="euler_1.html#E16:28_10"><i><font color="Green" title="E52">A137</font></i></a>, <a class="txt" href="euler_1.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a></i>;<br/>
<a NAME="E33:28_10"/><i><font color="Green" title="E62">A191</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c13">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c26">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E35:28_10"/><i><font color="Green" title="E63">A192</font></i>: 
<font color="Maroon" title="c26">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">y</font>,<font color="Maroon" title="c24">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/>
<a NAME="E36:28_10"/><i><font color="Green" title="E64">A193</font></i>: 
<font color="Maroon" title="c26">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">f</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E33:28_10"><i><font color="Green" title="E62">A191</font></i></a>, <a class="txt" href="euler_1.html#E35:28_10"><i><font color="Green" title="E63">A192</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E37:28_10"/>
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c25">y</font>,<font color="Maroon" title="c24">x</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E32:28_10"><i><font color="Green" title="E61">A190</font></i></a>, <a class="txt" href="euler_1.html#E23:28"><i><font color="Green" title="E41">A73</font></i></a></i>;<br/>
<b>hence </b><a NAME="E38:28_10"/>
<font color="Maroon" title="c16">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">f</font>
 <b>by </b><i><a class="txt" href="euler_1.html#E4:28_10"><i><font color="Green" title="E43">A128</font></i></a>, <a class="txt" href="euler_1.html#E36:28_10"><i><font color="Green" title="E64">A193</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="txt" href="euler_1.html#E35:28_10"><i><font color="Green" title="E63">A192</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:28"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c12">A</font>,<font color="Maroon" title="c13">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c14">C</font>
 
<b>by </b><i><a class="txt" href="euler_1.html#E24:28"><i><font color="Green" title="E42">A74</font></i></a>, <a class="ref" href="euler_1.html#T12" target="_self" title="EULER_1:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E27:28"/>
 <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c4">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="euler_1.html#E14:28"><i><font color="Green" title="E36">A40</font></i></a>, <a class="ref" href="card_2.html#T65" title="CARD_2:th.65">CARD_2:65</a></i>;<br/>


</div>
