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<b>let </b><font color="Maroon" title="c1">G</font> be   <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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>





<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>( <font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font> implies  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">h</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> $1 );<br/>
<a NAME="E1:71"/><i><font color="Green" title="E56">A1</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">k</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:71_1"/><i><font color="Green" title="E57">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 ;<br/>

<b>assume </b><a NAME="E2:71_1"/><i><font color="Green" title="E58">A3</font></i>: 
<font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_1_1"/>
( <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_1"><b>suppose </b></a><a NAME="E1:71_1_1_1"/>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:71_1_1_1"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">h</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T6" title="NAT_D:th.6">NAT_D:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_2"><b>suppose </b></a><a NAME="E1:71_1_1_2"/><i><font color="Green" title="E59">A4</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_1_1_2_1"/>
(  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:71_1_1_2_1_1"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:71_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c2">h</font> <a href="group_1.html#NR2" title="GROUP_1:NR.2">is_of_order_0</a> 
 <b>by </b><i><a class="ref" href="group_1.html#D12" target="_self" title="GROUP_1:def.12">Def12</a></i>;<br/><b>hence </b><a NAME="E3:71_1_1_2_1_1"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">h</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:71_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="group_1.html#E1:71_1_1_2"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="group_1.html#D11" target="_self" title="GROUP_1:def.11">Def11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:71_1_1_2_1_2"/><i><font color="Green" title="E60">A5</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">h</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:71_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c2">h</font> <a href="group_1.html#NR1" title="GROUP_1:NR.1">is_not_of_order_0</a> 
 <b>by </b><i><a class="ref" href="group_1.html#D12" target="_self" title="GROUP_1:def.12">Def12</a></i>;<br/><a NAME="E3:71_1_1_2_1_2"/><b>then </b>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">h</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:71_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="group_1.html#E1:71_1_1_2"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="group_1.html#D12" target="_self" title="GROUP_1:def.12">Def12</a></i>;<br/><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="E5:71_1_1_2_1_2"/><i><font color="Green" title="E61">A6</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">h</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c4">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">h</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">m</font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E5:71_1_1_2_1_2"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="group_1.html#T48" target="_self" title="GROUP_1:th.48">Th48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">m</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T82" target="_self" title="GROUP_1:th.82">Th82</a></i>
<br/>.= 
<font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">m</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" target="_self" title="GROUP_1:def.5">Def5</a></i>
;<br/><a NAME="E7:71_1_1_2_1_2"/><b>then </b>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">h</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="group_1.html#E1:71_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="group_1.html#E2:71_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="group_1.html#E1:71_1_1_2_1_2"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="group_1.html#E5:71_1_1_2_1_2"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="nat_1.html#T16" title="NAT_1:th.16">NAT_1:16</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:71_1_1_2_1_2"/><i><font color="Green" title="E62">A7</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">h</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><a NAME="E10:71_1_1_2_1_2"/>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">h</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_1.html#E5:71_1_1_2_1_2"><i><font color="Green" title="E61">A6</font></i></a>, <a class="txt" href="group_1.html#E9:71_1_1_2_1_2"><i><font color="Green" title="E62">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E11:71_1_1_2_1_2"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">h</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:71"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="group_1.html#E1:71"><i><font color="Green" title="E56">A1</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:71"/>
( <font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font> implies  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c2">h</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">n</font> )
 ;<br/>


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