<?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="E1:16"/><i><font color="Green">E15</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="E2:16"/><i><font color="Green">E16</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/>

<a NAME="E3:16"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16"><i><font color="Green">E15</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:16"><i><font color="Green">E16</font></i></a>, <a class="ref" href="group_1.html#T90">GROUP_1:90</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</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="E5:16"/><i><font color="Green">E18</font></i>: 
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>ex <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#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Olive">b<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16"><i><font color="Green">E15</font></i></a>, <a class="ref" href="gr_cy_1.html#T42">GR_CY_1:42</a></i>;<br/>
<a NAME="E6:16"/><i><font color="Green">E19</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>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_1"/>
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>ex <font color="Olive">b<sub>2</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:16_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E5:16"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E4:16_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</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#E2:16_1_1"><i><font color="Green">E20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:16_1"/>
<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>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="gr_cy_2.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:16"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="group_6.html#NM2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="group_6.html#NR3" target="_self">is_isomorphism</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:16_2"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E3:16_2"/><i><font color="Green">E21</font></i>: 
 <a href="gr_cy_1.html#K5">INT.Group</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="gr_cy_1.html#D6">GR_CY_1:def 6</a></i>;<br/>
<a NAME="E4:16_2"/><i><font color="Green">E22</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:16_2"/><i><font color="Green">E23</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1"/><i><font color="Green">E24</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E2:16_2_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_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> ;<br/>

<b>assume </b><a NAME="E1:16_2_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<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="E4:16_2_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E5:16"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E5:16_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:16_2_1_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</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#E3:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="nat_1.html#T46">NAT_1:46</a></i>;<br/>
<i><font color="Green">E27</font></i>: <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>3</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> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green">E17</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:16_2_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</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> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <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#K4">mod</a> <font color="Maroon">c<sub>1</sub></font></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>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>1</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> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <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#K4">mod</a> <font color="Maroon">c<sub>1</sub></font></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> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <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#K4">mod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16"><i><font color="Green">E15</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:16"><i><font color="Green">E16</font></i></a>, <a class="ref" href="gr_cy_1.html#T29">GR_CY_1:29</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>3</sub></font> <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#K4">mod</a> <font color="Maroon">c<sub>1</sub></font></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>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="gr_cy_2.html#E1:16_2_1_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E4:16_2_1_1_1_1"/><i><font color="Green">E28</font></i>: 
 <a href="gr_cy_2.html#K1" target="_self">@</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="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E5:16_2_1_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E2:16_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:16_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:16_2_1_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:16_2_1_1_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:16_2_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:16_2_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:16_2_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16_2_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:16_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2"><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="funct_1.html#D8">FUNCT_1:def 8</a><br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:16_2_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:16_2_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E3:16_2_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E4:16_2_2"/><i><font color="Green">E28</font></i>: 
<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>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16_2_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="gr_cy_2.html#E2:16_2_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E29</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> = 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:16_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:16_2_2"><i><font color="Green">E27</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:16_2"><i><font color="Green">E20</font></i></a></i>
;<br/>
<a NAME="E8:16_2_2"/>
ex <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 <br/>( <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> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>7</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<a NAME="E1:16_2_2_2"/><i><font color="Green">E30</font></i>: 
 <a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E2:16_2_2_2"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>7</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#E3:16"><i><font color="Green">E17</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:16_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<a NAME="E3:16_2_2_2"/><i><font color="Green">E32</font></i>: 
 <a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:16_2_2_2"/><b>then </b>
 <a href="gr_cy_2.html#K1" target="_self">@</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#E3:16"><i><font color="Green">E17</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:16_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/>
<b>hence </b><a NAME="E5:16_2_2_2"/>
ex <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 <br/>( <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> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</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#E4:16_2_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="gr_cy_2.html#E7:16_2_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="gr_cy_2.html#E1:16_2_2_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:16_2_2_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:16_2_2_2"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:16_2_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E11:16_2_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E12:16_2_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E13:16_2_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>
<a NAME="E14:16_2_2"/><i><font color="Green">E34</font></i>: 
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#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_2_3_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_1"><b>case </b></a><a NAME="E1:16_2_2_3_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16_2_2_3_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_2_3_1_1_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:16_2_2_3_1_1_1_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>14</sub></font> = 0;<br/><a NAME="E2:16_2_2_3_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E11:16_2_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="group_1.html#T63">GROUP_1:63</a></i>;<br/><a NAME="E3:16_2_2_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> 0
 <b>by </b><i><a class="ref" href="group_1.html#T63">GROUP_1:63</a></i>;<br/><a NAME="E4:16_2_2_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>13</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="ref" href="group_1.html#T43">GROUP_1:43</a></i>;<br/><a NAME="E5:16_2_2_3_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_3_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_2_3_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:16_2_2_3_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_2_3_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>13</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#E13:16_2_2"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_2_3_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:16_2_2_3_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E4:16_2_2_3_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="xcmplx_0.html#K2">+</a> 0
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E13:16_2_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:16_2_2_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E5:16_2_2_3_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E6:16_2_2_3_1_1_1_1_1_1"/>
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#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <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> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16_2_2_3_1_1_1_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:16_2_2_3_1_1_1_1_1_1"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_2_3_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:16_2_2_3_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><b>hence </b><a NAME="E3:16_2_2_3_1_1_1_1_1_2"/>
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#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <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> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E10:16_2_2"><i><font color="Green">E30</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="E5:16_2_2_3_1_1_1"/>
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#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_2"><b>case </b></a><a NAME="E1:16_2_2_3_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16_2_2_3_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_2_3_1_1_2_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:16_2_2_3_1_1_2_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>14</sub></font> = 0;<br/><a NAME="E2:16_2_2_3_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E11:16_2_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="group_1.html#T63">GROUP_1:63</a></i>;<br/><a NAME="E3:16_2_2_3_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> 0
 <b>by </b><i><a class="ref" href="group_1.html#T63">GROUP_1:63</a></i>;<br/><a NAME="E4:16_2_2_3_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>13</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="ref" href="group_1.html#T43">GROUP_1:43</a></i>;<br/><a NAME="E5:16_2_2_3_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_3_1_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_3_1_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_2_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_2_3_1_1_2_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_2_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:16_2_2_3_1_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_2_3_1_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>13</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#E12:16_2_2"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_2_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_2_3_1_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:16_2_2_3_1_1_2_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xcmplx_0.html#K4">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="int_1.html#NM1">Integer</a> ;<br/><a NAME="E4:16_2_2_3_1_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="xcmplx_0.html#K2">+</a> 0
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E12:16_2_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:16_2_2_3_1_1_2_1_1_1_1_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E5:16_2_2_3_1_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:16_2_2_3_1_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:16_2_2_3_1_1_2_1_1_1"/>
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#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <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> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16_2_2_3_1_1_2_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:16_2_2_3_1_1_2_1_1_1"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_2_3_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:16_2_2_3_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><b>hence </b><a NAME="E3:16_2_2_3_1_1_2_1_1_2"/>
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#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <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> )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E10:16_2_2"><i><font color="Green">E30</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="E5:16_2_2_3_1_1_2"/>
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#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <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> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2_3_1_1_3"><b>case </b></a><a NAME="E1:16_2_2_3_1_1_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_2_2_3_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E10:16_2_2"><i><font color="Green">E30</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="E2:16_2_2_3"/>
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#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <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> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E15:16_2_2"/><i><font color="Green">E35</font></i>: 
<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#E1:16"><i><font color="Green">E15</font></i></a>, <a class="ref" href="gr_cy_1.html#T26">GR_CY_1:26</a></i>;<br/>
<a NAME="E16:16_2_2"/>
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>3</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#E1:16"><i><font color="Green">E15</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:16"><i><font color="Green">E16</font></i></a>, <a class="txt" href="gr_cy_2.html#E6:16"><i><font color="Green">E19</font></i></a>, <a class="ref" href="gr_cy_1.html#T27">GR_CY_1:27</a></i>;<br/>
<b>hence </b><a NAME="E17:16_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E14:16_2_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="gr_cy_2.html#E15:16_2_2"><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>
<a NAME="E7:16_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</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="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_3"><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="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<a NAME="E1:16_2_3"/><i><font color="Green">E25</font></i>: 
 <a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1">Segm</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16_2"><i><font color="Green">E21</font></i></a></i>;<br/>
 <a href="gr_cy_2.html#K1" target="_self">@</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4">addint</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="gr_cy_1.html#D5">GR_CY_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16_2_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="gr_cy_2.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="newton.html#T77">NEWTON:77</a></i>
;<br/>
<b>then </b><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</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"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:16_2"><i><font color="Green">E20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T64">GROUP_1:64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p0">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T71">GROUP_1:71</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</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="group_1.html#K6">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K3">"</a> </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"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p5">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:16"><i><font color="Green">E15</font></i></a>, <a class="txt" href="gr_cy_2.html#E2:16"><i><font color="Green">E16</font></i></a>, <a class="ref" href="gr_cy_1.html#T29">GR_CY_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <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#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T42">GROUP_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T16">GROUP_1:16</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></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"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><a href="gr_cy_2.html#K1" target="_self">@</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:16_2"><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="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</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="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E2:16_2"><i><font color="Green">E20</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:16_2_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</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="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="group_6.html#NM2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/>
<a NAME="E9:16_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a> 
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E5:16_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="group_6.html#D13">GROUP_6:def 13</a></i>;<br/>
<a NAME="E10:16_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="group_6.html#NR1" target="_self">is_monomorphism</a> 
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E6:16_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="group_6.html#D12">GROUP_6:def 12</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E11:16_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="group_6.html#NR3" target="_self">is_isomorphism</a> 
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E9:16_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="gr_cy_2.html#E10:16_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="group_6.html#D14">GROUP_6:def 14</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:16"/>
 <a href="gr_cy_1.html#K5">INT.Group</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="group_6.html#R2">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="group_6.html#D15">GROUP_6:def 15</a></i>;<br/>


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