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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">G</font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:12"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a>
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">G</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">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green">E50</font></i>: 
 <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E4:12"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="group_1.html#T90">GROUP_1:90</a></i>;<br/>
<a NAME="E5:12"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">G</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"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</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">E62</font></i>: 
<font color="Maroon">G</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"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E2:12_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<b>set </b><font color="Maroon">d</font> = <font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font>;<br/>
<a NAME="E3:12_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E4:12_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">l</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:12_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:12_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E9:12_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<i><font color="Green">E71</font></i>: <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font> = 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#T50">GROUP_1:50</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">l</font>
<b>by </b><i><a class="ref" href="group_1.html#T50">GROUP_1:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">l</font>
<b>by </b><i>, , <a class="ref" href="gr_cy_1.html#T29">GR_CY_1:29</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
<b>by </b><i><a class="ref" href="group_1.html#T42">GROUP_1:42</a></i>
;<br/>
<a NAME="E11:12_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, , , <a class="ref" href="group_1.html#T90">GROUP_1:90</a></i>;<br/>
<a NAME="E12:12_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">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">E82</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>
<a NAME="E2:12_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">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">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:12_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> 1
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:12_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T5">XCMPLX_1:5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:12_1_2"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:12_1"/>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font> <a href="group_1.html#NR1" target="_self">is_not_of_order_0</a> 
 
<b>by </b><i>, <a class="ref" href="gr_cy_1.html#T26">GR_CY_1:26</a></i>;<br/>
<a NAME="E14:12_1"/><b>then </b>
 <a href="group_1.html#K7">ord</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, , <a class="ref" href="group_1.html#D12">GROUP_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E15:12_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="gr_cy_1.html#T27">GR_CY_1:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:12"/>
( <font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">G</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"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font></span>)</span></span><a href="domain_1.html#K6">}</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">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being   <a href="int_1.html#NM1">Integer</a>, <font color="Maroon">i1</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:12_2"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="newton.html#T81">NEWTON:81</a></i>;<br/>
<a NAME="E4:12_2"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <font color="Olive">b</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"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font></span>)</span></span><a href="domain_1.html#K6">}</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">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>

<a NAME="E1:12_2_1"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">i0</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:12_2_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">i0</font>
 <b>by </b><i>, <a class="ref" href="gr_cy_1.html#T25">GR_CY_1:25</a></i>;<br/>
<i><font color="Green">E85</font></i>: <font color="Maroon">i0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i0</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i0</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i0</font></span>)</span>

;<br/>
<a NAME="E5:12_2_1"/>
 ex <font color="Olive">i2</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Olive">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">E86</font></i>: <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i0</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i0</font></span>)</span></span>)</span></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"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i0</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i0</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T68">GROUP_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i0</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i0</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="gr_cy_1.html#T29">GR_CY_1:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i0</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T61">GROUP_1:61</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i0</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i0</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T68">GROUP_1:68</a></i>
;<br/>
<b>take </b>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i0</font>
;<br/>

<b>thus </b><a NAME="E2:12_2_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i0</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:12_2_1"/>
<font color="Maroon">b</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"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T25">GR_CY_1:25</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:12"/>
( <font color="Maroon">G</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"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> iff <font color="Maroon">k</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i/>;<br/>


</div>
