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

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



<b>assume </b><a NAME="E1:24"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">a</font>,<font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a>  )
 ;<br/>

<a NAME="E2:24"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  ex <font color="Olive">u</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Olive">u</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  ) </span>}</span> ;<br/>
<a NAME="E3:24"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  holds <br/><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Olive">x</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:24_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:24_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">m</font>,<font color="Maroon">t</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E5:24_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 ;<br/>
<a NAME="E6:24_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E7:24_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">t</font>,<font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E8:24_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E9:24_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font>,<font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i>, , <a class="ref" href="euler_1.html#T15">EULER_1:15</a></i>;<br/>
<a NAME="E10:24_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E11:24_1"/><b>then </b>
<font color="Maroon">t</font>,<font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<a NAME="E12:24_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:24_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">s</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:24_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
 <b>and </b><br/><a NAME="E4:24_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<a NAME="E5:24_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:24_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="nat_1.html#K10">hcf</a> 1 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T64">NEWTON:64</a></i>;<br/>
<a NAME="E7:24_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="euler_1.html#T6">EULER_1:6</a>, </i>;<br/>
<b>hence </b><a NAME="E8:24_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:24_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E15:24_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E16:24_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<b>hence </b><a NAME="E17:24_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span> 
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:24"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">xi</font>, <font color="Olive">xj</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">xi</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  &amp; <font color="Olive">xj</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  &amp; <font color="Olive">xi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">xj</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Olive">xi</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Olive">xj</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:24_2"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">xi</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  &amp; <font color="Maroon">xj</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  &amp; <font color="Maroon">xi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">xj</font> )
 ;<br/>

<b>assume </b><a NAME="E2:24_2"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">xi</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">xj</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
 ;<br/>

<b>set </b><font color="Maroon">tm</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">xi</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">sm</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">xj</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>;<br/>
<a NAME="E3:24_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:24_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xi</font>
 <b>and </b><br/><a NAME="E6:24_2"/>
<font color="Maroon">m</font>,<font color="Maroon">t</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E7:24_2"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i/>;<br/>
<a NAME="E8:24_2"/><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">xi</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span> <a href="nat_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="newton.html#T77">NEWTON:77</a></i>;<br/>
<b>consider </b><font color="Maroon">s</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:24_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xj</font>
 <b>and </b><br/><a NAME="E11:24_2"/>
<font color="Maroon">m</font>,<font color="Maroon">s</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E12:24_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">s</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i/>;<br/>
<a NAME="E13:24_2"/><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">xj</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="newton.html#T77">NEWTON:77</a></i>;<br/>
<a NAME="E14:24_2"/>
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span> <a href="nat_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E15:24_2"/><b>then </b>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">t</font></span>)</span> <a href="nat_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">s</font></span>)</span> <a href="nat_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E16:24_2"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">m</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#D9">INT_1:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font>, <font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">c</font> = <font color="Maroon">t</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E18:24_2"/>
<font color="Maroon">a</font>,<font color="Maroon">m</font> <a href="int_2.html#R1">are_relative_prime</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E19:24_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">m</font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="txt" href="euler_2.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="int_2.html#T40">INT_2:40</a></i>;<br/>
<a NAME="E20:24_2"/>
( 1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> 1 )
 
<b>by </b><i>, , <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E21:24_2"/><b>then </b>
<font color="Maroon">t</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="txt" href="euler_2.html#E2"><i><font color="Green">Lemma34</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E22:24_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:24"/><i><font color="Green">E111</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  ex <font color="Olive">u</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Olive">u</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:24_3"/>
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  ex <font color="Olive">u</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Olive">u</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">xx</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:24_3_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xx</font>
 <b>and </b><br/><a NAME="E4:24_3_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">m</font>,<font color="Maroon">xx</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E5:24_3_1"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">xx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">xx</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 ;<br/>
<a NAME="E6:24_3_1"/>
<font color="Maroon">xx</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font>
 
<b>by </b><i>, , <a class="ref" href="euler_1.html#T2">EULER_1:2</a></i>;<br/>
<a NAME="E7:24_3_1"/><b>then </b>
<font color="Maroon">xx</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E8:24_3_1"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">xx</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xx</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">j</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E10:24_3_1"/><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">xx</font>
 <b>by </b><i>, <a class="ref" href="euler_1.html#T11">EULER_1:11</a></i>;<br/>
<a NAME="E11:24_3_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="newton.html#T80">NEWTON:80</a></i>;<br/>
<a NAME="E12:24_3_1"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E13:24_3_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E14:24_3_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_3_1_1"/>
<font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:24_3_1_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="int_1.html#D8">INT_1:def 8</a></i>;<br/>
<a NAME="E3:24_3_1_1"/><b>then </b><i><font color="Green">E121</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font>
 
;<br/>
<a NAME="E4:24_3_1_1"/><b>then </b>
<font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:24_3_1_1"/><b>then </b><i><font color="Green">E122</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i/>;<br/>
<font color="Maroon">m</font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">xx</font> = 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="int_2.html#K3">gcd</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="euler_2.html#E2"><i><font color="Green">Lemma34</font></i></a></i>
<br/>.= 
<font color="Maroon">m</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="int_2.html#K3">gcd</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="euler_1.html#T16">EULER_1:16</a></i>
<br/>.= 
<font color="Maroon">m</font> <a href="nat_1.html#K2">*</a> 1
<b>by </b><i/>
<br/>.= 
<font color="Maroon">m</font>

;<br/>
<a NAME="E7:24_3_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:24_3_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:24_3_1"/>
<font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T78">NEWTON:78</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">im</font> = <font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E17:24_3_1"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">im</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="euler_2.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E18:24_3_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">im</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="newton.html#T79">NEWTON:79</a></i>;<br/>
<b>reconsider </b><font color="Maroon">ij</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, </i>;<br/>
<i><font color="Green">E127</font></i>: <font color="Maroon">ij</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K5">div</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="int_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="euler_1.html#T13">EULER_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">im</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="newton.html#T101">NEWTON:101</a></i>
;<br/>
<a NAME="E21:24_3_1"/><b>then </b>
<font color="Maroon">m</font>,<font color="Maroon">im</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i>, , , , , <a class="txt" href="euler_2.html#E1"><i><font color="Green">Lemma33</font></i></a>, </i>;<br/>
<a NAME="E22:24_3_1"/><b>then </b>
<font color="Maroon">im</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="euler_2.html#E2"><i><font color="Green">Lemma34</font></i></a>, <a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E23:24_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  ex <font color="Olive">u</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Olive">u</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  ) </span>}</span> 
 <b>by </b><i>, , , <a class="txt" href="euler_2.html#E2:24"><i><font color="Green">E36</font></i></a></i>;<br/>


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

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

<b>assume </b><a NAME="E2:24_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  ex <font color="Olive">u</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Olive">u</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">yy</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:24_3"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yy</font>
 <b>and </b><br/><a NAME="E5:24_3"/><i><font color="Green">E130</font></i>: 
 ex <font color="Olive">u</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">yy</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Olive">u</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  )
 ;<br/>
<b>consider </b><font color="Maroon">uu</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:24_3"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">yy</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">uu</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E8:24_3"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">uu</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span> 
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E9:24_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span> 
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">FX</font> =  <a href="finseq_1.html#K14">Sgm</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">FYY</font> = <font color="Maroon">a</font> <a href="euler_2.html#K1" target="_self">*</a> <font color="Maroon">FX</font> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">FY</font> = <font color="Maroon">FYY</font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">m</font> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E9:24"/><i><font color="Green">E137</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">xx</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:24_4"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xx</font>
 <b>and </b><br/><a NAME="E4:24_4"/>
<font color="Maroon">m</font>,<font color="Maroon">xx</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E5:24_4"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">xx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">xx</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 ;<br/>
<b>thus </b><a NAME="E6:24_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<i><font color="Green">E140</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FX</font> = 
 <a href="card_1.html#K4">card</a> <font color="Maroon">X</font>
<b>by </b><i>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>
<br/>.= 
 <a href="euler_1.html#K1">Euler</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="euler_1.html#D1">EULER_1:def 1</a></i>
;<br/>
<i><font color="Green">E141</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FY</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FYY</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FX</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T6">NEWTON:6</a></i>
;<br/>
<a NAME="E13:24"/><b>then </b><i><font color="Green">E142</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FX</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FY</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FYY</font> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FYY</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FX</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T6">NEWTON:6</a></i>
;<br/>
<a NAME="E15:24"/><b>then </b><i><font color="Green">E143</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FX</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FYY</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E16:24"/><i><font color="Green">E144</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">FY</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  ex <font color="Olive">u</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Olive">u</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_8"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:24_8"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">FY</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  ex <font color="Olive">u</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Olive">u</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_8_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">FY</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_8_1"/><i><font color="Green">E145</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FY</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FY</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:24_8_1"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E6:24_8_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FX</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E147</font></i>: <font color="Maroon">FY</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">FYY</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">y</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>
;<br/>
<a NAME="E8:24_8_1"/>
<font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">FX</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:24_8_1"/><b>then </b>
<font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E10:24_8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  ex <font color="Olive">u</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Olive">u</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  ) </span>}</span> 
 <b>by </b><i>, <a class="txt" href="euler_2.html#E1"><i><font color="Green">Lemma33</font></i></a></i>;<br/>


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

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

<b>assume </b><a NAME="E2:24_8"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  ex <font color="Olive">u</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Olive">u</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Maroon">m</font>,<font color="Olive">k</font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span>  ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">yy</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:24_8"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yy</font>
 <b>and </b><br/><a NAME="E5:24_8"/><i><font color="Green">E149</font></i>: 
 ex <font color="Olive">u</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">yy</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Olive">u</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> &amp; <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>
<b>consider </b><font color="Maroon">uu</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:24_8"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">yy</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">uu</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E8:24_8"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">uu</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>
<a NAME="E9:24_8"/>
<font color="Maroon">uu</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">FX</font>
 
<b>by </b><i>, <a class="txt" href="euler_2.html#E2"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">xx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:24_8"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FX</font> &amp; <font color="Maroon">uu</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FX</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:24_8"/>
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FX</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">xx</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<font color="Maroon">FY</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">xx</font> = 
<span class="p1">(<span class="default"><font color="Maroon">FYY</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">xx</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E35</font></i></a>, </i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="txt" href="euler_2.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>
;<br/>
<b>hence </b><a NAME="E15:24_8"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">FY</font>
 <b>by </b><i>, <a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:24"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">FY</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:24_9"/><i><font color="Green">E156</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FY</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FY</font> &amp; <font color="Maroon">FY</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FY</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<a NAME="E2:24_9"/><b>then </b><i><font color="Green">E157</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FX</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FX</font> &amp; <font color="Maroon">FY</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FY</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<i><font color="Green">E158</font></i>: <font color="Maroon">FY</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">FYY</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="txt" href="euler_2.html#E1"><i><font color="Green">Lemma33</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>
;<br/>
<i><font color="Green">E159</font></i>: <font color="Maroon">FY</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">FYY</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x2</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="txt" href="euler_2.html#E1"><i><font color="Green">Lemma33</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>
;<br/>
<a NAME="E7:24_9"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">FX</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:24_9"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">FX</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:24_9"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">FX</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T99">FINSEQ_3:99</a></i>;<br/>
<a NAME="E10:24_9"/>
(  not <font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  not <font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  not <font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">x2</font> )
 
<b>by </b><i>, , <a class="txt" href="euler_2.html#E2"><i><font color="Green">Lemma34</font></i></a>, <a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:24_9"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, <a class="txt" href="euler_2.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="euler_2.html#E2:24"><i><font color="Green">E36</font></i></a>, <a class="txt" href="euler_2.html#E3"><i><font color="Green">Lemma37</font></i></a>, <a class="txt" href="euler_2.html#E4"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:24"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds   <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">FY</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_10_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_10_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_10_1_1_1"><b>suppose </b></a><a NAME="E1:24_10_1_1_1"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:24_10_1_1_1"/><b>then </b><i><font color="Green">E164</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">FX</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E3:24_10_1_1_1"/>
<font color="Maroon">FX</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T99">FINSEQ_3:99</a></i>;<br/><a NAME="E4:24_10_1_1_1"/><b>then </b><i><font color="Green">E165</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FX</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="euler_2.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="finseq_3.html#T97">FINSEQ_3:97</a></i>;<br/><a NAME="E5:24_10_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FX</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">FX</font> <a href="relat_1.html#K10">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T66">FINSEQ_3:66</a></i>;<br/><a NAME="E6:24_10_1_1_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FX</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">FX</font> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 ;<br/><a NAME="E7:24_10_1_1_1"/><b>then </b><i><font color="Green">E166</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FX</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 ;<br/><a NAME="E8:24_10_1_1_1"/><i><font color="Green">E167</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, , , , <a class="ref" href="finseq_3.html#T97">FINSEQ_3:97</a></i>;<br/><a NAME="E9:24_10_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">FY</font> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T66">FINSEQ_3:66</a></i>;<br/><a NAME="E10:24_10_1_1_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">FY</font> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_3.html#K1">-</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p3">(<span class="default"><font color="Maroon">FY</font> <a href="relat_1.html#K10">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">FY</font> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E11:24_10_1_1_1"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">FY</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="euler_2.html#E2"><i><font color="Green">Lemma34</font></i></a>, <a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="euler_2.html#E2:24"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_10_1_1_2"><b>suppose </b></a><a NAME="E1:24_10_1_1_2"/><i><font color="Green">E168</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:24_10_1_1_2"/><b>then </b><i><font color="Green">E169</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">FX</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E3:24_10_1_1_2"/>
<font color="Maroon">FY</font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T142">FUNCT_1:142</a></i>;<br/><b>hence </b><a NAME="E4:24_10_1_1_2"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">FY</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="euler_2.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="ref" href="funct_1.html#T142">FUNCT_1:142</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:24_10"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">FY</font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E19:24"/><b>then </b><i><font color="Green">E170</font></i>: 
<font color="Maroon">FX</font>,<font color="Maroon">FY</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="rfinseq.html#D1">RFINSEQ:def 1</a></i>;<br/>
<a NAME="E20:24"/><b>then </b><i><font color="Green">E171</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">FX</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">FY</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E21:24"/><i><font color="Green">E172</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">FX</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font>
 
<b>by </b><i>, <a class="ref" href="rfinseq.html#T16">RFINSEQ:16</a></i>;<br/>
<a NAME="E22:24"/><i><font color="Green">E173</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">FY</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">FX</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_11"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font> implies <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p3">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> );<br/>
<a NAME="E1:24_11"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_11_1"><b>proof </b></a><div class="add">

<a NAME="E1:24_11_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font>
 
;<br/>
<a NAME="E2:24_11_1"/><b>then </b><i><font color="Green">E175</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E3:24_11_1"/>
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T124">RVSUM_1:124</a></i>;<br/>
<a NAME="E4:24_11_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> 0</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_2.html#E3"><i><font color="Green">Lemma37</font></i></a>, <a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/>
<b>hence </b><a NAME="E5:24_11_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="ref" href="rvsum_1.html#T124">RVSUM_1:124</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:24_11"/><i><font color="Green">E176</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_11_2"><b>proof </b></a><div class="add">

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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_11_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_11_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_11_2_1_1"/>
( <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font> or <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_11_2_1_1_1"><b>suppose </b></a><a NAME="E1:24_11_2_1_1_1"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:24_11_2_1_1_1"/><i><font color="Green">E178</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><a NAME="E3:24_11_2_1_1_1"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:24_11_2_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font>
 <b>by </b><i><a class="txt" href="euler_2.html#E4"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:24_11_2_1_1_1"/><b>then </b><i><font color="Green">E180</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E6:24_11_2_1_1_1"/><i><font color="Green">E181</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="euler_2.html#E4"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E7:24_11_2_1_1_1"/><b>then </b><i><font color="Green">E182</font></i>: 
<font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="rfinseq.html#T20">RFINSEQ:20</a></i>;<br/><a NAME="E8:24_11_2_1_1_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:24_11_2_1_1_1"/><b>then </b><i><font color="Green">E183</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FY</font>
 <b>by </b><i><a class="txt" href="euler_2.html#E4"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E10:24_11_2_1_1_1"/><i><font color="Green">E184</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="euler_2.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_5.html#T80">FINSEQ_5:80</a></i>;<br/><a NAME="E11:24_11_2_1_1_1"/><i><font color="Green">E185</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">FY</font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a></i>;<br/><a NAME="E12:24_11_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="euler_2.html#E2"><i><font color="Green">Lemma34</font></i></a>, <a class="txt" href="euler_2.html#E4"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E13:24_11_2_1_1_1"/><b>then </b><i><font color="Green">E186</font></i>: 
<font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="rfinseq.html#T20">RFINSEQ:20</a></i>;<br/><a NAME="E14:24_11_2_1_1_1"/><i><font color="Green">E187</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="euler_2.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_5.html#T80">FINSEQ_5:80</a></i>;<br/><a NAME="E15:24_11_2_1_1_1"/><i><font color="Green">E188</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a></i>;<br/><span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p3">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">FY</font> <a href="wsierp_1.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p5">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">FY</font> <a href="wsierp_1.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i><a class="txt" href="euler_2.html#E4"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="euler_2.html#T1" target="_self">Th1</a>, <a class="ref" href="euler_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p5">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">FYY</font> <a href="wsierp_1.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p5">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="euler_2.html#K1" target="_self">*</a> <font color="Maroon">FX</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p5">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="euler_2.html#K1" target="_self">*</a> <font color="Maroon">FX</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a>, <a class="txt" href="euler_2.html#E7"><i><font color="Green">Lemma41</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p4">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i><a class="txt" href="euler_2.html#E3:24"><i><font color="Green">E42</font></i></a>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p3">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="txt" href="euler_2.html#E8"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="euler_2.html#T5" target="_self">Th5</a>, </i>
;<br/><b>hence </b><a NAME="E17:24_11_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_11_2_1_1_2"><b>suppose </b></a><a NAME="E1:24_11_2_1_1_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_11_2_1_1_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1] )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:24_11_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:24_11"/><i><font color="Green">E189</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="euler_2.html#E2:24"><i><font color="Green">E36</font></i></a>, <a class="txt" href="euler_2.html#E3"><i><font color="Green">Lemma37</font></i></a>);<br/></i>
<a NAME="E4:24_11"/>
<font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">FY</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E5:24_11"/><b>then </b><i><font color="Green">E190</font></i>: 
<font color="Maroon">FY</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FY</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T23">FINSEQ_2:23</a></i>;<br/>
<a NAME="E6:24_11"/>
<font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FX</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">FX</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FX</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E7:24_11"/><b>then </b>
<font color="Maroon">FX</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_2.html#E2"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="finseq_2.html#T23">FINSEQ_2:23</a></i>;<br/>
<b>hence </b><a NAME="E8:24_11"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">FY</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">FX</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="euler_2.html#E4"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="euler_2.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:24"/><i><font color="Green">E191</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">FY</font> <a href="hidden.html#R1">=</a>  <a href="euler_1.html#K1">Euler</a> <font color="Maroon">m</font>
 
<b>by </b><i>, , <a class="ref" href="rfinseq.html#T16">RFINSEQ:16</a></i>;<br/>
<a NAME="E24:24"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><a href="euler_1.html#K1">Euler</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="prepower.html#T12">PREPOWER:12</a></i>;<br/>
<a NAME="E25:24"/><i><font color="Green">E193</font></i>: 
 <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">FX</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_12"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_12"/>
 <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">FX</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:24_12"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FX</font>
 <b>and </b><br/><a NAME="E4:24_12"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="rvsum_1.html#T133">RVSUM_1:133</a></i>;<br/>
<a NAME="E5:24_12"/><i><font color="Green">E196</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">FX</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/>
<a NAME="E6:24_12"/>
<font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">FX</font>
 
<b>by </b><i><a class="ref" href="euler_2.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:24_12"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E9:24_12"/>
<font color="Maroon">m</font>,<font color="Maroon">p</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E10:24_12"/><i><font color="Green">E198</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="txt" href="euler_2.html#E7"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<b>thus </b><a NAME="E11:24_12"/>
contradiction
 <b>by </b><i><a class="ref" href="euler_2.html#T2" target="_self">Th2</a>, <a class="txt" href="euler_2.html#E3:24"><i><font color="Green">E42</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:24"/>
 <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">FX</font>,<font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_13"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FX</font> implies  <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a>  );<br/>
<a NAME="E1:24_13"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_13_1"><b>proof </b></a><div class="add">

<a NAME="E1:24_13_1"/>
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T124">RVSUM_1:124</a></i>;<br/>
<a NAME="E2:24_13_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <span class="p2">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> 0</span>)</span></span>)</span> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T64">NEWTON:64</a></i>;<br/>
<b>hence </b><a NAME="E3:24_13_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:24_13"/><i><font color="Green">E200</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_13_2"><b>proof </b></a><div class="add">

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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_13_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_13_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_13_2_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FX</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">FX</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_13_2_1_1_1"><b>suppose </b></a><a NAME="E1:24_13_2_1_1_1"/><i><font color="Green">E201</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">FX</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>assume </b><a NAME="E2:24_13_2_1_1_1"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><a NAME="E3:24_13_2_1_1_1"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>reconsider </b><font color="Maroon">FF</font> = <font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><b>reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">FF</font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E6:24_13_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">FF</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="euler_2.html#E7"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E7:24_13_2_1_1_1"/><b>then </b><i><font color="Green">E206</font></i>: 
<font color="Maroon">FF</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">ff</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="rfinseq.html#T20">RFINSEQ:20</a></i>;<br/><a NAME="E8:24_13_2_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E9:24_13_2_1_1_1"/><b>then </b><i><font color="Green">E207</font></i>: 
<font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T80">FINSEQ_5:80</a></i>;<br/><a NAME="E10:24_13_2_1_1_1"/><i><font color="Green">E208</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">FX</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E11:24_13_2_1_1_1"/><i><font color="Green">E209</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">FX</font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T121">FINSEQ_3:121</a></i>;<br/><a NAME="E12:24_13_2_1_1_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E13:24_13_2_1_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FX</font>
 <b>by </b><i><a class="txt" href="euler_2.html#E7"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:24_13_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">FX</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E16:24_13_2_1_1_1"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">ff</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E17:24_13_2_1_1_1"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">m</font>,<font color="Maroon">p</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E18:24_13_2_1_1_1"/>
( <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">a</font> =  <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><font color="Maroon">FF</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">b</font> = <font color="Maroon">ff</font>, <font color="Maroon">m'</font> = <font color="Maroon">m</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E20:24_13_2_1_1_1"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">m'</font> <a href="int_2.html#R1">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="euler_2.html#E7"><i><font color="Green">Lemma41</font></i></a>, <a class="txt" href="euler_2.html#E3:24"><i><font color="Green">E42</font></i></a>, , , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E21:24_13_2_1_1_1"/>
<font color="Maroon">b</font>,<font color="Maroon">m'</font> <a href="int_2.html#R1">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="euler_2.html#E8"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="euler_2.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E22:24_13_2_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">b</font>,<font color="Maroon">m'</font> <a href="int_2.html#R1">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="euler_2.html#T6" target="_self">Th6</a>, <a class="ref" href="int_2.html#T41">INT_2:41</a></i>;<br/><a NAME="E23:24_13_2_1_1_1"/><b>then </b>
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FF</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">ff</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<font color="Maroon">m'</font> <a href="int_2.html#R1">are_relative_prime</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E24:24_13_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_13_2_1_1_2"><b>suppose </b></a><a NAME="E1:24_13_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">FX</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_13_2_1_1_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1] )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:24_13_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:24_13"/><i><font color="Green">E214</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="euler_2.html#T1" target="_self">Th1</a>, <a class="ref" href="euler_2.html#T2" target="_self">Th2</a>);<br/></i>
<a NAME="E4:24_13"/>
<font color="Maroon">FX</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">FX</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">FX</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/>
<b>hence </b><a NAME="E5:24_13"/>
 <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">FX</font>,<font color="Maroon">m</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="euler_2.html#E7"><i><font color="Green">Lemma41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:24"/><b>then </b>
<font color="Maroon">m</font>, <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">FX</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
;<br/>
<b>hence </b><a NAME="E28:24"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><a href="euler_1.html#K1">Euler</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="txt" href="euler_2.html#E1"><i><font color="Green">Lemma33</font></i></a>, <a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="euler_2.html#E2:24"><i><font color="Green">E36</font></i></a>, <a class="txt" href="euler_2.html#E3"><i><font color="Green">Lemma37</font></i></a>, <a class="txt" href="euler_2.html#E4"><i><font color="Green">Lemma38</font></i></a>, </i>;<br/>


</div>
