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

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">H</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1"/>
( <font color="Maroon" title="c2">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a>  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c1">G</font> or <font color="Maroon" title="c2">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c1">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><b>suppose </b></a><a NAME="E1:25_1_1"/>
<font color="Maroon" title="c2">H</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a>  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c1">G</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1"/>
<font color="Maroon" title="c2">H</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2"><b>suppose </b></a><a NAME="E1:25_1_2"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c2">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c1">G</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c3">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:25_1_2"/><i><font color="Green" title="E15">A2</font></i>: 
 <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#D9" title="GR_CY_1:def.9">GR_CY_1:def 9</a></i>;<br/><b>consider </b><font color="Maroon" title="c4">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><a NAME="E5:25_1_2"/>
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">m</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> &amp; ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">j</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> holds <br/><font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_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" title="NAT_1:NM.1">Nat</a>] <b>means </b>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> $1 &amp; <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> $1 <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> );<br/>
<a NAME="E1:25_1_2_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>by </b><i><a class="txt" href="group_8.html#E1:25_1_2"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="group_8.html#E3:25_1_2"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="group_8.html#T24" target="_self" title="GROUP_8:th.24">Th24</a></i>;<br/>
<a NAME="E2:25_1_2_1"/><b>then </b><i><font color="Green" title="E16">A3</font></i>: 
 ex <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:25_1_2_1"/>
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] &amp; ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">j</font>] holds <br/><font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> ) )
 
<b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="group_8.html#E2:25_1_2_1"><i><font color="Green" title="E16">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:25_1_2_1"/>
 ex <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> &amp; <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">m</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> &amp; ( for <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">j</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> holds <br/><font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> ) )
 ;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c5">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:25_1_2"/><i><font color="Green" title="E16">A4</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">m</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> &amp; ( for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">j</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> holds <br/><font color="Maroon" title="c5">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> ) )
 ;<br/><b>set </b><font color="Maroon" title="c6">c</font> = <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">m</font>;<br/><b>reconsider </b><font color="Maroon" title="c7">c</font> = <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">m</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font> <b>by </b><i><a class="txt" href="group_8.html#E7:25_1_2"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E9:25_1_2"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font>  ex <font color="Olive" title="b2">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b2">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font>;<br/>

<a NAME="E1:25_1_2_2"/>
 ex <font color="Olive" title="b1">t</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">t</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:25_1_2_2_1"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T50" title="GROUP_2:th.50">GROUP_2:50</a></i>;<br/>
<a NAME="E2:25_1_2_2_1"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="group_8.html#E3:25_1_2"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="group_2.html#T50" title="GROUP_2:th.50">GROUP_2:50</a></i>;<br/>
<a NAME="E3:25_1_2_2_1"/>
<font color="Maroon" title="c8">a</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="group_8.html#E1:25_1_2_2_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">j1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E5:25_1_2_2_1"/><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">j1</font>
 <b>by </b><i><a class="txt" href="group_8.html#E2:25_1_2_2_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="gr_cy_1.html#T25" title="GR_CY_1:th.25">GR_CY_1:25</a></i>;<br/>
<a NAME="E6:25_1_2_2_1"/>
<font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">j1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="txt" href="group_8.html#E5:25_1_2_2_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:25_1_2_2_1"/>
 ex <font color="Olive" title="b1">t</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">t</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> )
 <b>by </b><i><a class="txt" href="group_8.html#E5:25_1_2_2_1"><i><font color="Green" title="E19">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c9">t</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:25_1_2_2"/><i><font color="Green" title="E17">A8</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">t</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> )
 ;<br/>
<a NAME="E4:25_1_2_2"/>
 ex <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_2_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c10">r</font> = <font color="Maroon" title="c9">t</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c5">m</font>;<br/>

<b>take </b><font color="Maroon" title="c11">s</font> = <font color="Maroon" title="c9">t</font> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <font color="Maroon" title="c5">m</font>;<br/>

<b>thus </b><a NAME="E1:25_1_2_2_2"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">s</font> &amp; <font color="Maroon" title="c11">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">s</font> )
 <b>by </b><i><a class="txt" href="group_8.html#E7:25_1_2"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="newton.html#T78" title="NEWTON:th.78">NEWTON:78</a>, <a class="ref" href="newton.html#T79" title="NEWTON:th.79">NEWTON:79</a>, <a class="ref" href="newton.html#T80" title="NEWTON:th.80">NEWTON:80</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c10">r</font>, <font color="Maroon" title="c11">s</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E6:25_1_2_2"/><i><font color="Green" title="E18">A9</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">s</font> &amp; <font color="Maroon" title="c11">s</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">m</font> &amp; <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">s</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">s</font> = <font color="Maroon" title="c11">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="group_8.html#E6:25_1_2_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<i><font color="Green" title="E19">A10</font></i>: <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">t</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">r</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">s</font></span>)</span>
<b>by </b><i><a class="txt" href="group_8.html#E6:25_1_2_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c10">r</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">s</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T67" title="GROUP_1:th.67">GROUP_1:67</a></i>
;<br/>
<a NAME="E9:25_1_2_2"/><i><font color="Green" title="E20">A11</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c10">r</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="txt" href="group_8.html#E7:25_1_2"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="group_8.html#T22" target="_self" title="GROUP_8:th.22">Th22</a></i>;<br/>
<a NAME="E10:25_1_2_2"/><i><font color="Green" title="E21">A12</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">s</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_2_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c13">u</font> = <font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">t</font>;<br/>
<b>set </b><font color="Maroon" title="c14">v</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c10">r</font>;<br/>
<i><font color="Green" title="E22">A13</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c10">r</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">t</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" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c10">r</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c10">r</font></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">s</font></span>)</span>
<b>by </b><i><a class="txt" href="group_8.html#E8:25_1_2_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">s</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">s</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/>
<a NAME="E2:25_1_2_2_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c10">r</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="txt" href="group_8.html#E9:25_1_2_2"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="group_2.html#T60" title="GROUP_2:th.60">GROUP_2:60</a></i>;<br/>
<b>hence </b><a NAME="E3:25_1_2_2_4"/>
<font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">s</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="txt" href="group_8.html#E3:25_1_2_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="group_8.html#E1:25_1_2_2_4"><i><font color="Green" title="E22">A13</font></i></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:25_1_2_2"/>
<font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_2_5"><b>proof </b></a><div class="add">

<a NAME="E1:25_1_2_2_5"/>
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="group_8.html#E7:25_1_2"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="group_8.html#E6:25_1_2_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="group_8.html#E10:25_1_2_2"><i><font color="Green" title="E21">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:25_1_2_2_5"/>
<font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="group_1.html#T43" title="GROUP_1:th.43">GROUP_1:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:25_1_2_2"/><b>then </b><i><font color="Green" title="E22">A14</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c10">r</font>
 
<b>by </b><i><a class="txt" href="group_8.html#E8:25_1_2_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>;<br/>
<a NAME="E13:25_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c10">r</font>
 
<b>by </b><i><a class="ref" href="group_4.html#T4" title="GROUP_4:th.4">GROUP_4:4</a></i>;<br/>
<b>hence </b><a NAME="E14:25_1_2_2"/>
 ex <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">i</font>
 <b>by </b><i><a class="txt" href="group_8.html#E3:25_1_2_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="group_8.html#E12:25_1_2_2"><i><font color="Green" title="E22">A14</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:25_1_2"/><b>then </b>
<font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">c</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="gr_cy_2.html#T11" title="GR_CY_2:th.11">GR_CY_2:11</a></i>;<br/><b>hence </b><a NAME="E11:25_1_2"/>
<font color="Maroon" title="c2">H</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>
 <b>by </b><i><a class="ref" href="gr_cy_2.html#T10" title="GR_CY_2:th.10">GR_CY_2:10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
