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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>;<br/>





<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c2">G</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">q</font> =  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font>;<br/>
<b>set </b><font color="Maroon" title="c5">p'</font> = <font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span>;<br/>
<a NAME="E2:17"/>
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:17"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/>
<a NAME="E3:17"/><b>then </b><i><font color="Green" title="E20">A2</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
;<br/>
<a NAME="E4:17"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">q'</font> = <span class="p1">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">q'</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T48" title="GROUP_1:th.48">GROUP_1:48</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">q'</font></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E3:17"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T64" title="GROUP_1:th.64">GROUP_1:64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p3">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T65" title="GROUP_1:th.65">GROUP_1:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p5">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T71" title="GROUP_1:th.71">GROUP_1:71</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p5">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c2">G</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">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p4">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T50" title="GROUP_1:th.50">GROUP_1:50</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="c2">G</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:17"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="gr_cy_1.html#T29" target="_self" title="GR_CY_1:th.29">Th29</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c2">G</font>
<b>by </b><i><a class="ref" href="group_1.html#T42" title="GROUP_1:th.42">GROUP_1:42</a></i>
;<br/>
<b>hence </b><a NAME="E7:17"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="group_1.html#T20" title="GROUP_1:th.20">GROUP_1:20</a></i>;<br/>


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