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

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c3">G</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:12"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c3">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c3">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="c4">a</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green" title="E13">A3</font></i>: 
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E4:12"/><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:12"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:12"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/>
<a NAME="E5:12"/><i><font color="Green" title="E15">A5</font></i>: 
( <font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c2">k</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies <font color="Maroon" title="c2">k</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:12_1"/><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c2">k</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E2:12_1"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/>

<b>set </b><font color="Maroon" title="c5">d</font> = <font color="Maroon" title="c2">k</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E3:12_1"/><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">k</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<a NAME="E4:12_1"/><i><font color="Green" title="E19">A9</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">n</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:12_1"/><i><font color="Green" title="E20">A10</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">l</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:12_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">p</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E8:12_1"/><i><font color="Green" title="E21">A11</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E4:12_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<a NAME="E9:12_1"/><i><font color="Green" title="E22">A12</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E8:12_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<i><font color="Green" title="E23">A13</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c7">p</font> = 
<font color="Maroon" title="c4">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"><font color="Maroon" title="c6">l</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p</font></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E6:12_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="group_1.html#T50" title="GROUP_1:th.50">GROUP_1:50</a></i>
<br/>.= 
<font color="Maroon" title="c4">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#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">l</font></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E8:12_1"><i><font color="Green" title="E21">A11</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">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#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">l</font>
<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="c3">G</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">l</font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:12"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:12"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="gr_cy_1.html#T29" title="GR_CY_1:th.29">GR_CY_1:29</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c3">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/>
<a NAME="E11:12_1"/><i><font color="Green" title="E24">A14</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:12"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:12"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="gr_cy_2.html#E8:12_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="group_1.html#T90" title="GROUP_1:th.90">GROUP_1:90</a></i>;<br/>
<a NAME="E12:12_1"/><i><font color="Green" title="E25">A15</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:12_1_2"/><i><font color="Green" title="E26">A16</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E4:12"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="gr_cy_2.html#E9:12_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/>
<a NAME="E2:12_1_2"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1_2_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E2:12_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> 1
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E8:12_1"><i><font color="Green" title="E21">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:12_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E2:12_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="gr_cy_2.html#E11:12_1"><i><font color="Green" title="E24">A14</font></i></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:12_1_2"/>
<font color="Maroon" title="c7">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:12_1_2"><i><font color="Green" title="E26">A16</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:12_1"/>
<font color="Maroon" title="c4">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c2">k</font> <a href="group_1.html#NR1" title="GROUP_1:NR.1">is_not_of_order_0</a> 
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:12"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="gr_cy_1.html#T26" title="GR_CY_1:th.26">GR_CY_1:26</a></i>;<br/>
<a NAME="E14:12_1"/><b>then </b>
 <a href="group_1.html#K7" title="GROUP_1:func.7">ord</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">p</font>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E10:12_1"><i><font color="Green" title="E23">A13</font></i></a>, <a class="txt" href="gr_cy_2.html#E11:12_1"><i><font color="Green" title="E24">A14</font></i></a>, <a class="ref" href="group_1.html#D12" title="GROUP_1:def.12">GROUP_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E15:12_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:12"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:12"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="gr_cy_2.html#E1:12_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="gr_cy_2.html#E12:12_1"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="gr_cy_1.html#T27" title="GR_CY_1:th.27">GR_CY_1:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:12"/>
( <font color="Maroon" title="c2">k</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c2">k</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_2"/>
<font color="Maroon" title="c2">k</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c6">i1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:12_2"/><i><font color="Green" title="E16">A17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E4:12"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="newton.html#T81" title="NEWTON:th.81">NEWTON:81</a></i>;<br/>
<a NAME="E4:12_2"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font> holds  <font color="Olive" title="b1">b</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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c2">k</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G</font>;<br/>

<a NAME="E1:12_2_1"/>
<font color="Maroon" title="c7">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">G</font>
 
<b>by </b><i><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="c8">i0</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:12_2_1"/><i><font color="Green" title="E17">A18</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c8">i0</font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E2:12"><i><font color="Green" title="E12">A2</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/>
<i><font color="Green" title="E18">A19</font></i>: <font color="Maroon" title="c8">i0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">i1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i0</font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:12_2"><i><font color="Green" title="E16">A17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i0</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">i1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i0</font></span>)</span>

;<br/>
<a NAME="E5:12_2_1"/>
 ex <font color="Olive" title="b1">i2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Olive" title="b1">i2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1_2"><b>proof </b></a><div class="add">

<i><font color="Green" title="E19">A20</font></i>: <font color="Maroon" title="c7">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i0</font></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="c4">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i0</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:12_2_1"><i><font color="Green" title="E17">A18</font></i></a>, <a class="txt" href="gr_cy_2.html#E4:12_2_1"><i><font color="Green" title="E18">A19</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"><font color="Maroon" title="c4">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"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i0</font></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="c4">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#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i0</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T68" title="GROUP_1:th.68">GROUP_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i0</font></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"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">i1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i0</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E1:12"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="gr_cy_2.html#E3:12"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="gr_cy_1.html#T29" title="GR_CY_1:th.29">GR_CY_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">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"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i0</font></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="c3">G</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T61" title="GROUP_1:th.61">GROUP_1:61</a></i>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i0</font></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="c4">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i0</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T68" title="GROUP_1:th.68">GROUP_1:68</a></i>
;<br/>
<b>take </b>
<font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i0</font>
;<br/>

<b>thus </b><a NAME="E2:12_2_1_2"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">i0</font></span>)</span>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E1:12_2_1_2"><i><font color="Green" title="E19">A20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:12_2_1"/>
<font color="Maroon" title="c7">b</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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c2">k</font></span>)</span></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#T25" title="GR_CY_1:th.25">GR_CY_1:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:12_2"/>
<font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c2">k</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="group_2.html#T71" title="GROUP_2:th.71">GROUP_2:71</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:12"/>
( <font color="Maroon" title="c3">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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c2">k</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> iff <font color="Maroon" title="c2">k</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E5:12"><i><font color="Green" title="E15">A5</font></i></a></i>;<br/>


</div>
