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

<b>let </b><font color="Maroon">c<sub>1</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><a NAME="E1:28"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:28"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">c<sub>1</sub></font>
 ;<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>1</sub></font><b> such that </b><br/><a NAME="E4:28"/><i><font color="Green">E20</font></i>: 
 <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">c<sub>1</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_1.html#D9">GR_CY_1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="int_1.html#NM1">Integer</a>;<br/>
<a NAME="E6:28"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<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="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</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>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">a<sub>1</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> );<br/>
<a NAME="E1:28_1"/><i><font color="Green">E21</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="group_8.html#E1:28"><i><font color="Green">E18</font></i></a>, <a class="txt" href="group_8.html#E2:28"><i><font color="Green">E19</font></i></a>, <a class="txt" href="group_8.html#E4:28"><i><font color="Green">E20</font></i></a>, <a class="ref" href="group_8.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E2:28_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="group_8.html#E1:28_1"><i><font color="Green">E21</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:28_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<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="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</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>2</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>then 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="E8:28"/><i><font color="Green">E21</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>5</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<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="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <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> <b>by </b><i><a class="txt" href="group_8.html#E8:28"><i><font color="Green">E21</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:28"/><i><font color="Green">E22</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="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>7</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="28_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E1:28_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <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="rlvect_1.html#R1">in</a> <font color="Maroon">c<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>8</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_2_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="group_2.html#T50">GROUP_2:50</a></i>;<br/>
<a NAME="E2:28_2_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="rlvect_1.html#R1">in</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="txt" href="group_8.html#E4:28"><i><font color="Green">E20</font></i></a></i>;<br/>
<a NAME="E3:28_2_1"/>
<font color="Maroon">c<sub>8</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E1:28_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E4:28_2_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">c<sub>8</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>1</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E2:28_2_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="gr_cy_1.html#T25">GR_CY_1:25</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E6:28_2_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>8</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>9</sub></font>
 ;<br/>
<a NAME="E7:28_2_1"/>
<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="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E6:28_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:28_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( <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="rlvect_1.html#R1">in</a> <font color="Maroon">c<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>8</sub></font> )
 <b>by </b><i><a class="txt" href="group_8.html#E6:28_2_1"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:28_2"/><i><font color="Green">E23</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="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <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>8</sub></font> )
 ;<br/>
<a NAME="E4:28_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</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>5</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="int_1.html#K5">div</a> <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>5</sub></font>;<br/>

<a NAME="E1:28_2_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E8:28"><i><font color="Green">E21</font></i></a>, <a class="ref" href="newton.html#T80">NEWTON:80</a></i>;<br/>
<a NAME="E2:28_2_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T78">NEWTON:78</a></i>;<br/>
<a NAME="E3:28_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="int_1.html#K6">mod</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E8:28"><i><font color="Green">E21</font></i></a>, <a class="ref" href="newton.html#T79">NEWTON:79</a></i>;<br/>
<b>hence </b><a NAME="E4:28_2_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="group_8.html#E1:28_2_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="group_8.html#E2:28_2_2"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E6:28_2"/><i><font color="Green">E24</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <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="group_8.html#E6:28_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<i><font color="Green">E25</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> = 
<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>5</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="group_8.html#E6:28_2"><i><font color="Green">E24</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"><font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>10</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> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T63">GROUP_1:63</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>5</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>10</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> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T67">GROUP_1:67</a></i>
;<br/>
<a NAME="E9:28_2"/><i><font color="Green">E26</font></i>: 
<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>5</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E8:28"><i><font color="Green">E21</font></i></a>, <a class="ref" href="group_8.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E10:28_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>9</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <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>5</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>10</sub></font>;<br/>
<a NAME="E1:28_2_4"/><i><font color="Green">E28</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> <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>5</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>10</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> <font color="Maroon">c<sub>12</sub></font></span>)</span> &amp; <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="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <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>5</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="group_8.html#E3:28_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="group_8.html#E8:28_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="group_8.html#E9:28_2"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>then </b><i><font color="Green">E29</font></i>: <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>5</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_1.html#K3">"</a> </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> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K10">*</a> <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>5</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>10</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> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</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> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<a NAME="E3:28_2_4"/>
<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>5</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="group_1.html#K3">"</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E1:28_2_4"><i><font color="Green">E28</font></i></a>, <a class="ref" href="group_2.html#T60">GROUP_2:60</a></i>;<br/>
<b>hence </b><a NAME="E4:28_2_4"/>
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>12</sub></font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="group_8.html#E1:28_2_4"><i><font color="Green">E28</font></i></a>, <a class="txt" href="group_8.html#E2:28_2_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:28_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_5"><b>proof </b></a><div class="add">

<a NAME="E1:28_2_5"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="group_8.html#E8:28"><i><font color="Green">E21</font></i></a>, <a class="txt" href="group_8.html#E6:28_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="group_8.html#E10:28_2"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:28_2_5"/>
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="group_1.html#T43">GROUP_1:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:28_2"/><b>then </b>
<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> <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>5</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E8:28_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>;<br/>
<a NAME="E13:28_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <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>5</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="group_8.html#E3:28_2"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E14:28_2"/>
<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>5</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="group_8.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E15:28_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K6">|^</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="group_8.html#E13:28_2"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:28"/>
<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>7</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="group_8.html#E10:28"><i><font color="Green">E22</font></i></a>, <a class="ref" href="gr_cy_2.html#T11">GR_CY_2:11</a></i>;<br/>
<b>hence </b><a NAME="E12:28"/>
<font color="Maroon">c<sub>2</sub></font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a>
 <b>by </b><i><a class="ref" href="gr_cy_2.html#T10">GR_CY_2:10</a></i>;<br/>


</div>
