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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  )
 ;<br/>

<a NAME="E2:24"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  ) </span>}</span> ;<br/>
<a NAME="E3:24"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:24_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:24_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:24_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E5:24_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>
<a NAME="E6:24_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_2.html#E4:24_1"><i><font color="Green">E24</font></i></a>, <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">E26</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">E27</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="euler_2.html#E5:24_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E9:24_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="euler_2.html#E7:24_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="euler_1.html#T15">EULER_1:15</a></i>;<br/>
<a NAME="E10:24_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="euler_2.html#E4:24_1"><i><font color="Green">E24</font></i></a>, <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">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></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">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="euler_2.html#E8:24_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="euler_2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E13:24_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></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">E30</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</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">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="euler_2.html#E2:24"><i><font color="Green">E21</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<a NAME="E5:24_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K8">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="E6:24_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="euler_1.html#T6">EULER_1:6</a></i>;<br/>
<b>hence </b><a NAME="E7:24_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="euler_2.html#E9:24_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="euler_2.html#E3:24_1_1"><i><font color="Green">E30</font></i></a>, <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">E30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="euler_2.html#E2:24"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E16:24_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="euler_2.html#E3:24_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="euler_2.html#E12:24_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="euler_2.html#E14:24_1"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:24"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:24_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E2:24_2"/><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E3:24_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="euler_2.html#E2:24"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:24_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E6:24_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E7:24_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="euler_2.html#E1:24_2"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E8:24_2"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_2.html#E3:24_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="euler_2.html#E5:24_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="newton.html#T77">NEWTON:77</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:24_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E11:24_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E12:24_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="euler_2.html#E1:24_2"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E13:24_2"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_2.html#E3:24_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="euler_2.html#E10:24_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="newton.html#T77">NEWTON:77</a></i>;<br/>
0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euler_2.html#E2:24_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="euler_2.html#E8:24_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="euler_2.html#E13:24_2"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E15:24_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E16:24_2"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>10</sub></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">c<sub>11</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E18:24_2"/>
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="int_2.html#R1">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E20</font></i></a>, <a class="ref" href="euler_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E19:24_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="int_1.html#R2">divides</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="euler_2.html#E16:24_2"><i><font color="Green">E33</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">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 )
 
<b>by </b><i><a class="txt" href="euler_2.html#E7:24_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="euler_2.html#E12:24_2"><i><font color="Green">E31</font></i></a>, <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">c<sub>9</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="euler_2.html#E19:24_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="euler_2.html#E7"><i><font color="Green">Lemma7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E22:24_2"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_2.html#E1:24_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="euler_2.html#E5:24_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="euler_2.html#E10:24_2"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:24"/><i><font color="Green">E24</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></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">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></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">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:24_3_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E4:24_3_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></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">E27</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>
<a NAME="E6:24_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="euler_2.html#E4:24_3_1"><i><font color="Green">E26</font></i></a>, <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">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="euler_2.html#E5:24_3_1"><i><font color="Green">E27</font></i></a>, <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">E28</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></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">E29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E20</font></i></a>, <a class="ref" href="euler_1.html#T11">EULER_1:11</a></i>;<br/>
<a NAME="E11:24_3_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="int_1.html#K5">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_2.html#E2:24"><i><font color="Green">E21</font></i></a>, <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">E31</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></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">c<sub>7</sub></font> <a href="int_1.html#K5">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_2.html#E10:24_3_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E13:24_3_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="euler_2.html#E2:24"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E14:24_3_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="int_1.html#K5">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_2.html#E2:24"><i><font color="Green">E21</font></i></a>, <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">E34</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="int_1.html#K5">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E4:24_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font> <a href="int_1.html#K5">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_2.html#E10:24_3_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E5:24_3_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="int_1.html#K5">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>8</sub></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><a class="txt" href="euler_2.html#E5:24_3_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#K3">gcd</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>7</sub></font> <a href="int_1.html#K5">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euler_2.html#E10:24_3_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="euler_2.html#E3:24_3_1_1"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font> <a href="int_1.html#K5">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="euler_2.html#E13:24_3_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="euler_2.html#E5:24_3_1_1"><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">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> 1
<b>by </b><i><a class="txt" href="euler_2.html#E4"><i><font color="Green">Lemma4</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>

;<br/>
<a NAME="E7:24_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="euler_2.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:24_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="euler_2.html#E4:24_3_1"><i><font color="Green">E26</font></i></a>, <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">c<sub>7</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="nat_1.html#NM1">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">E34</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="euler_2.html#E14:24_3_1"><i><font color="Green">E33</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">E35</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="euler_2.html#E13:24_3_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="newton.html#T79">NEWTON:79</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></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">c<sub>7</sub></font> <a href="int_1.html#K5">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="euler_2.html#E10:24_3_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="euler_2.html#E11:24_3_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<i><font color="Green">E36</font></i>: <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></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">c<sub>7</sub></font> <a href="int_1.html#K5">div</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>9</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="euler_2.html#E4:24_3_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="euler_2.html#E8:24_3_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="euler_2.html#E10:24_3_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="euler_2.html#E11:24_3_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="euler_2.html#E14:24_3_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="euler_2.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E22:24_3_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="euler_2.html#E17:24_3_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="euler_2.html#E18:24_3_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E23:24_3_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="euler_2.html#E3:24_3_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="euler_2.html#E8:24_3_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="euler_2.html#E12:24_3_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="euler_2.html#E20:24_3_1"><i><font color="Green">E36</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:24_3"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E5:24_3"/><i><font color="Green">E26</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:24_3"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E8:24_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="euler_2.html#E5:24_3"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>thus </b><a NAME="E9:24_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="euler_2.html#E3:24"><i><font color="Green">E22</font></i></a>, <a class="txt" href="euler_2.html#E4:24_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="euler_2.html#E7:24_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="euler_2.html#E8:24_3"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="finseq_1.html#K14">Sgm</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></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">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="euler_2.html#K2" target="_self">mod</a> <font color="Maroon">c<sub>2</sub></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">E25</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></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">c<sub>8</sub></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">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:24_4"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E4:24_4"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E5:24_4"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E6:24_4"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="euler_2.html#E3:24_4"><i><font color="Green">E26</font></i></a>, <a class="txt" href="euler_2.html#E5:24_4"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></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">E26</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="euler_2.html#E9:24"><i><font color="Green">E25</font></i></a>, <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">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="euler_1.html#D1">EULER_1:def 1</a></i>
;<br/>
<i><font color="Green">E27</font></i>:  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></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">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euler_2.html#D1" target="_self">Def1</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">c<sub>5</sub></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">E28</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>5</sub></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">E29</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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">E30</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>9</sub></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">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_8_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></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">E32</font></i>: 
<font color="Maroon">c<sub>10</sub></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">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_2.html#E3:24_8_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="euler_2.html#E4:24_8_1"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E6:24_8_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="euler_2.html#E12:24"><i><font color="Green">E27</font></i></a>, <a class="txt" href="euler_2.html#E3:24_8_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E33</font></i>: <font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="euler_2.html#E15:24"><i><font color="Green">E29</font></i></a>, <a class="ref" href="euler_2.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="euler_2.html#E13:24"><i><font color="Green">E28</font></i></a>, <a class="txt" href="euler_2.html#E3:24_8_1"><i><font color="Green">E31</font></i></a>, <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">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="euler_2.html#E9:24"><i><font color="Green">E25</font></i></a>, <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">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="euler_2.html#E3:24_8_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="euler_2.html#E7:24_8_1"><i><font color="Green">E33</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:24_8"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E5:24_8"/><i><font color="Green">E32</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:24_8"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E8:24_8"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="euler_2.html#E5:24_8"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E9:24_8"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="euler_2.html#E9:24"><i><font color="Green">E25</font></i></a>, <a class="txt" href="euler_2.html#E8:24_8"><i><font color="Green">E34</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">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:24_8"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></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">c<sub>12</sub></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">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_2.html#E11:24_8"><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">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="euler_2.html#E15:24"><i><font color="Green">E29</font></i></a>, <a class="txt" href="euler_2.html#E11:24_8"><i><font color="Green">E35</font></i></a>, <a class="ref" href="euler_2.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="euler_2.html#E4:24_8"><i><font color="Green">E31</font></i></a>, <a class="txt" href="euler_2.html#E7:24_8"><i><font color="Green">E33</font></i></a>, <a class="txt" href="euler_2.html#E11:24_8"><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">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="euler_2.html#E13:24"><i><font color="Green">E28</font></i></a>, <a class="txt" href="euler_2.html#E11:24_8"><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">E31</font></i>: 
<font color="Maroon">c<sub>7</sub></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">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></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">E32</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/>

<a NAME="E2:24_9"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="euler_2.html#E12:24"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="euler_2.html#E1:24_9"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="euler_2.html#E1:24_9"><i><font color="Green">E32</font></i></a></i>;<br/>
<i><font color="Green">E34</font></i>: <font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="euler_2.html#E15:24"><i><font color="Green">E29</font></i></a>, <a class="txt" href="euler_2.html#E2:24_9"><i><font color="Green">E33</font></i></a>, <a class="ref" href="euler_2.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>
;<br/>
<i><font color="Green">E35</font></i>: <font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="euler_2.html#E15:24"><i><font color="Green">E29</font></i></a>, <a class="txt" href="euler_2.html#E2:24_9"><i><font color="Green">E33</font></i></a>, <a class="ref" href="euler_2.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></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">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="euler_2.html#E13:24"><i><font color="Green">E28</font></i></a>, <a class="txt" href="euler_2.html#E1:24_9"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:24_9"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="euler_2.html#E13:24"><i><font color="Green">E28</font></i></a>, <a class="txt" href="euler_2.html#E1:24_9"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:24_9"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="euler_2.html#E9:24"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_3.html#T99">FINSEQ_3:99</a></i>;<br/>
<a NAME="E10:24_9"/>
(  not <font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="euler_2.html#E4:24"><i><font color="Green">E23</font></i></a>, <a class="txt" href="euler_2.html#E1:24_9"><i><font color="Green">E32</font></i></a>, <a class="txt" href="euler_2.html#E5:24_9"><i><font color="Green">E34</font></i></a>, <a class="txt" href="euler_2.html#E6:24_9"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:24_9"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="euler_2.html#E9:24"><i><font color="Green">E25</font></i></a>, <a class="txt" href="euler_2.html#E2:24_9"><i><font color="Green">E33</font></i></a>, <a class="txt" href="euler_2.html#E7:24_9"><i><font color="Green">E36</font></i></a>, <a class="txt" href="euler_2.html#E8:24_9"><i><font color="Green">E37</font></i></a>, <a class="txt" href="euler_2.html#E9:24_9"><i><font color="Green">E38</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">b<sub>1</sub></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">c<sub>5</sub></font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>7</sub></font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>9</sub></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">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> or  not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></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">E32</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:24_10_1_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="euler_2.html#E9:24"><i><font color="Green">E25</font></i></a>, <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">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="euler_2.html#E9:24"><i><font color="Green">E25</font></i></a>, <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">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="euler_2.html#E2:24_10_1_1_1"><i><font color="Green">E33</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">c<sub>5</sub></font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></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">c<sub>5</sub></font> <a href="relat_1.html#K10">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="finseq_3.html#K1">-</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></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">c<sub>5</sub></font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">E35</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></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">c<sub>5</sub></font> <a href="finseq_3.html#K1">-</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">E36</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="euler_2.html#E5:24"><i><font color="Green">E24</font></i></a>, <a class="txt" href="euler_2.html#E16:24"><i><font color="Green">E30</font></i></a>, <a class="txt" href="euler_2.html#E17:24"><i><font color="Green">E31</font></i></a>, <a class="txt" href="euler_2.html#E1:24_10_1_1_1"><i><font color="Green">E32</font></i></a>, <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">c<sub>7</sub></font> <a href="finseq_3.html#K1">-</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></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">c<sub>7</sub></font> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></font> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></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">c<sub>7</sub></font> <a href="finseq_3.html#K1">-</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></font> <a href="relat_1.html#K10">"</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></font> <a href="relat_1.html#K10">"</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="euler_2.html#E4:24_10_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="euler_2.html#E7:24_10_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="euler_2.html#E8:24_10_1_1_1"><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">E32</font></i>: 
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:24_10_1_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="euler_2.html#E9:24"><i><font color="Green">E25</font></i></a>, <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">c<sub>7</sub></font> <a href="relat_1.html#K10">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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="txt" href="euler_2.html#E5:24"><i><font color="Green">E24</font></i></a>, <a class="txt" href="euler_2.html#E16:24"><i><font color="Green">E30</font></i></a>, <a class="txt" href="euler_2.html#E1:24_10_1_1_2"><i><font color="Green">E32</font></i></a>, <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">c<sub>5</sub></font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="euler_2.html#E2:24_10_1_1_2"><i><font color="Green">E33</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">c<sub>5</sub></font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></font> <a href="relat_1.html#K10">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>9</sub></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">E32</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></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">E33</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="euler_2.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E21:24"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="euler_2.html#E19:24"><i><font color="Green">E32</font></i></a>, <a class="ref" href="rfinseq.html#T16">RFINSEQ:16</a></i>;<br/>
<a NAME="E22:24"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>7</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></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="nat_1.html#NM1">Nat</a>] means ( <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">c<sub>7</sub></font> implies <span class="p1">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></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#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>7</sub></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#K9">Product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></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#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> );<br/>
<a NAME="E1:24_11"/><i><font color="Green">E36</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">c<sub>7</sub></font>
 
;<br/>
<a NAME="E2:24_11_1"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></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#K9">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></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">c<sub>7</sub></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#K9">Product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></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#E2:24_11_1"><i><font color="Green">E37</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">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">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">c<sub>9</sub></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">c<sub>7</sub></font> or <font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></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">E38</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></font>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:24_11_2_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>]
 ;<br/><a NAME="E3:24_11_2_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="euler_2.html#E1:24_11_2_1_1_1"><i><font color="Green">E38</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">E41</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></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">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="euler_2.html#E1:24_11_2_1_1_1"><i><font color="Green">E38</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">E43</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">E44</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="euler_2.html#E1:24_11_2_1_1_1"><i><font color="Green">E38</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">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="euler_2.html#E3:24_11_2_1_1_1"><i><font color="Green">E40</font></i></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">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="euler_2.html#E21:24"><i><font color="Green">E34</font></i></a>, <a class="txt" href="euler_2.html#E1:24_11_2_1_1_1"><i><font color="Green">E38</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">E47</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">E48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="euler_2.html#E3:24_11_2_1_1_1"><i><font color="Green">E40</font></i></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">E49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></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#K9">Product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="euler_2.html#E7:24_11_2_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="euler_2.html#E10:24_11_2_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="euler_2.html#E11:24_11_2_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="euler_2.html#E18"><i><font color="Green">Lemma14</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></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#K9">Product</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="euler_2.html#E1:24_11_2_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="euler_2.html#E2:24_11_2_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="euler_2.html#E3:24_11_2_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="euler_2.html#T11" target="_self">Th11</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">c<sub>1</sub></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">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></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#K9">Product</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="euler_2.html#E13:24"><i><font color="Green">E28</font></i></a>, <a class="txt" href="euler_2.html#E15:24"><i><font color="Green">E29</font></i></a>, <a class="txt" href="euler_2.html#E9:24_11_2_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="euler_2.html#D1" target="_self">Def1</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">c<sub>1</sub></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">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></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#K9">Product</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="euler_2.html#T7" target="_self">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></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#K9">Product</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font> <a href="euler_2.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="wsierp_1.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="euler_2.html#T11" target="_self">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></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#K9">Product</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="euler_2.html#E5:24_11_2_1_1_1"><i><font color="Green">E41</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">c<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>1</sub></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#K9">Product</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></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#K9">Product</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="euler_2.html#E6:24_11_2_1_1_1"><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">c<sub>1</sub></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">c<sub>7</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>9</sub></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#K9">Product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="euler_2.html#E13:24_11_2_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="euler_2.html#E14:24_11_2_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="euler_2.html#E15:24_11_2_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="euler_2.html#E18"><i><font color="Green">Lemma14</font></i></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>7</sub></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">c<sub>9</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:24_11"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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#E1:24_11"><i><font color="Green">E36</font></i></a>, <a class="txt" href="euler_2.html#E2:24_11"><i><font color="Green">E37</font></i></a>);<br/></i>
<a NAME="E4:24_11"/>
<font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></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">E39</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></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">c<sub>7</sub></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">c<sub>5</sub></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">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></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">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></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">c<sub>7</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="euler_2.html#E21:24"><i><font color="Green">E34</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#K9">Product</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>7</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="euler_2.html#E3:24_11"><i><font color="Green">E38</font></i></a>, <a class="txt" href="euler_2.html#E5:24_11"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:24"/><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="euler_1.html#K1">Euler</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="euler_2.html#E11:24"><i><font color="Green">E26</font></i></a>, <a class="txt" href="euler_2.html#E19:24"><i><font color="Green">E32</font></i></a>, <a class="ref" href="rfinseq.html#T16">RFINSEQ:16</a></i>;<br/>
<a NAME="E24:24"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E20</font></i></a>, <a class="ref" href="prepower.html#T12">PREPOWER:12</a></i>;<br/>
<a NAME="E25:24"/><i><font color="Green">E38</font></i>: 
 <a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>5</sub></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#K9">Product</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:24_12"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:24_12"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>9</sub></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">E41</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="euler_2.html#E9:24"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/>
<a NAME="E6:24_12"/>
<font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="euler_2.html#E3:24_12"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:24_12"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E9:24_12"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>and </b><br/><a NAME="E10:24_12"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="euler_2.html#E5:24_12"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>thus </b><a NAME="E11:24_12"/>
contradiction
 <b>by </b><i><a class="txt" href="euler_2.html#E4:24_12"><i><font color="Green">E40</font></i></a>, <a class="txt" href="euler_2.html#E8:24_12"><i><font color="Green">E42</font></i></a>, <a class="txt" href="euler_2.html#E10:24_12"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:24"/>
 <a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></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="nat_1.html#NM1">Nat</a>] means ( <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">c<sub>5</sub></font> implies  <a href="wsierp_1.html#K9">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a>  );<br/>
<a NAME="E1:24_13"/><i><font color="Green">E39</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#K9">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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#K9">Product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> 0</span>)</span></span>)</span> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>2</sub></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">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">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">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></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">E41</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>9</sub></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">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font>]
 ;<br/><a NAME="E3:24_13_2_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></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">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E6:24_13_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="euler_2.html#E1:24_13_2_1_1_1"><i><font color="Green">E41</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">E44</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></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">c<sub>11</sub></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">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></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">E45</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></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">E46</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="euler_2.html#E9:24"><i><font color="Green">E25</font></i></a>, <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">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="euler_2.html#E1:24_13_2_1_1_1"><i><font color="Green">E41</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">c<sub>5</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="euler_2.html#E11:24_13_2_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E16:24_13_2_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E17:24_13_2_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>12</sub></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">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="euler_2.html#E10:24_13_2_1_1_1"><i><font color="Green">E46</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> =  <a href="wsierp_1.html#K9">Product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E20:24_13_2_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="int_2.html#R1">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="euler_2.html#E1:24_13_2_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="euler_2.html#E2:24_13_2_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="euler_2.html#E3:24_13_2_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="euler_2.html#E9:24_13_2_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="euler_2.html#T1" target="_self">Th1</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">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="int_2.html#R1">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="euler_2.html#E16:24_13_2_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="euler_2.html#E17:24_13_2_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="euler_2.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E22:24_13_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="int_2.html#R1">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="euler_2.html#E20:24_13_2_1_1_1"><i><font color="Green">E50</font></i></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#K9">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></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">c<sub>11</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<font color="Maroon">c<sub>15</sub></font> <a href="int_2.html#R1">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="euler_2.html#E18"><i><font color="Green">Lemma14</font></i></a></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">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="euler_2.html#E7:24_13_2_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="euler_2.html#T1" target="_self">Th1</a></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">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:24_13"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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#E1:24_13"><i><font color="Green">E39</font></i></a>, <a class="txt" href="euler_2.html#E2:24_13"><i><font color="Green">E40</font></i></a>);<br/></i>
<a NAME="E4:24_13"/>
<font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></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#K9">Product</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="euler_2.html#E3:24_13"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:24"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>, <a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></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">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="euler_2.html#E1:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="euler_2.html#E20:24"><i><font color="Green">E33</font></i></a>, <a class="txt" href="euler_2.html#E22:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="euler_2.html#E23:24"><i><font color="Green">E36</font></i></a>, <a class="txt" href="euler_2.html#E24:24"><i><font color="Green">E37</font></i></a>, <a class="txt" href="euler_2.html#E25:24"><i><font color="Green">E38</font></i></a>, <a class="ref" href="euler_2.html#T27" target="_self">Th27</a></i>;<br/>


</div>
