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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="group_1.html#V1">strict</a> <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:22"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="group_1.html#V6">finite</a>
 <b>and </b><br/><a NAME="E2:22"/><i><font color="Green">E19</font></i>: 
 <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E3:22"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:22"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#D9">GR_CY_1:def 9</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font>
;<br/>

<i><font color="Green">E22</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> 2 = 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="group_1.html#T50">GROUP_1:50</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:22"><i><font color="Green">E18</font></i></a>, <a class="ref" href="gr_cy_1.html#T29">GR_CY_1:29</a></i>
;<br/>
<a NAME="E7:22"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#NR1" target="_self">is_not_of_order_0</a> 
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:22"><i><font color="Green">E18</font></i></a>, <a class="ref" href="gr_cy_1.html#T26">GR_CY_1:26</a></i>;<br/>
<a NAME="E8:22"/><i><font color="Green">E24</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:22"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="gr_cy_2.html#E5:22"><i><font color="Green">E21</font></i></a>, <a class="ref" href="gr_cy_1.html#T27">GR_CY_1:27</a></i>;<br/>
<a NAME="E9:22"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:22"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="group_1.html#T90">GROUP_1:90</a></i>;<br/>
<a NAME="E10:22"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E11:22"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 holds <br/>2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><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><b>that </b><br/><a NAME="E1:22_2"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:22_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>and </b><br/><a NAME="E3:22_2"/><i><font color="Green">E30</font></i>: 
2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E4:22_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:22"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gr_cy_2.html#E10:22"><i><font color="Green">E26</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:22_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E5:22_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#NR1" target="_self">is_not_of_order_0</a> 
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:22"><i><font color="Green">E18</font></i></a>, <a class="ref" href="gr_cy_1.html#T26">GR_CY_1:26</a></i>;<br/>
<a NAME="E6:22_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E9:22"><i><font color="Green">E25</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:22_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E7:22_2"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:22_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="group_1.html#T50">GROUP_1:50</a></i>;<br/>
<b>hence </b><a NAME="E8:22_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E8:22"><i><font color="Green">E24</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:22_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="gr_cy_2.html#E5:22_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="gr_cy_2.html#E6:22_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="group_1.html#D12">GROUP_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:22"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st  <a href="group_1.html#K7">ord</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:22_3"/><i><font color="Green">E28</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 2
 <b>and </b><br/><a NAME="E2:22_3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>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="E4:22_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:22"><i><font color="Green">E18</font></i></a>, <a class="txt" href="gr_cy_2.html#E5:22"><i><font color="Green">E21</font></i></a>, <a class="ref" href="gr_cy_2.html#T12" target="_self">Th12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:22_3_1"/><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"><font color="Maroon">c<sub>3</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> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E3:22_3_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E10:22"><i><font color="Green">E26</font></i></a>, <a class="ref" href="nat_1.html#T42">NAT_1:42</a></i>;<br/><a NAME="E4:22_3_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_3_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>then </b><i><font color="Green">E33</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K4">mod</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:22_3_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K4">mod</a> 2</span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K2">*</a> 2</span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K4">mod</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T48">GROUP_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K2">*</a> 2</span>)</span></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K4">mod</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T50">GROUP_1:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K3">div</a> 2</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K4">mod</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E6:22"><i><font color="Green">E22</font></i></a>, <a class="ref" href="group_1.html#T50">GROUP_1:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K4">mod</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T42">GROUP_1:42</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K4">mod</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_3_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_3_1_1_2_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T62">NAT_1:62</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_3_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:22_3_1_1_2_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:22_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E2:22_3_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="group_1.html#T43">GROUP_1:43</a></i>;<br/><b>hence </b><a NAME="E3:22_3_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:22_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:22_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="group_1.html#T84">GROUP_1:84</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_3_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:22_3_1_1_2_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_3_1_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E2:22_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:22_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:22_3_1_1"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:22_3_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div><a NAME="E5:22_3_1"/><b>then </b><i><font color="Green">E33</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E6:22_3_1"/><i><font color="Green">E34</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:22"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:22_3_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E7:22_3_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3_1_2"><b>proof </b></a><div class="add">

<b>thus </b> <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> 2
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:22_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:22_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="group_1.html#T82">GROUP_1:82</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</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> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:22_3_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="group_1.html#T50">GROUP_1:50</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><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>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:22"><i><font color="Green">E20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.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>7</sub></font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T48">GROUP_1:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p3">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E8:22"><i><font color="Green">E24</font></i></a>, <a class="ref" href="group_1.html#T50">GROUP_1:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T82">GROUP_1:82</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K2">*</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T42">GROUP_1:42</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E8:22_3_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#NR1" target="_self">is_not_of_order_0</a> 
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:22"><i><font color="Green">E18</font></i></a>, <a class="ref" href="gr_cy_1.html#T26">GR_CY_1:26</a></i>;<br/><b>hence </b><a NAME="E9:22_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E8:22"><i><font color="Green">E24</font></i></a>, <a class="txt" href="gr_cy_2.html#E5:22_3_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="gr_cy_2.html#E6:22_3_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="gr_cy_2.html#E7:22_3_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="group_1.html#D12">GROUP_1:def 12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:22_3"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:22"/>
(  <a href="group_1.html#K7">ord</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 2 &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st  <a href="group_1.html#K7">ord</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E6:22"><i><font color="Green">E22</font></i></a>, <a class="txt" href="gr_cy_2.html#E7:22"><i><font color="Green">E23</font></i></a>, <a class="txt" href="gr_cy_2.html#E11:22"><i><font color="Green">E27</font></i></a>, <a class="ref" href="group_1.html#D12">GROUP_1:def 12</a></i>;<br/>


</div>
