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<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">G</font> implies  <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">h</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E1:75"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:75_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 ;<br/>

<b>assume </b><a NAME="E2:75_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">G</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1"><b>suppose </b></a><a NAME="E1:75_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:75_1_1_1"/>
 <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">h</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#T53">NAT_1:53</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_2"><b>suppose </b></a><a NAME="E1:75_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:75_1_1_2_1_1"/>
(  <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 0 or  <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:75_1_1_2_1_1_1"/>
 <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:75_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">h</font> <a href="group_1.html#NR2" target="_self">is_of_order_0</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:75_1_1_2_1_1_1"/>
 <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">h</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:75"><i><font color="Green">E60</font></i></a>, , <a class="txt" href="group_1.html#E1:75_1_1_2"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:75_1_1_2_1_1_2"/><i><font color="Green">E55</font></i>: 
 <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:75_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">h</font> <a href="group_1.html#NR1" target="_self">is_not_of_order_0</a> 
 <b>by </b><i><a class="txt" href="group_1.html#E5:75_1_1_2_1_1_2"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E3:75_1_1_2_1_1_2"/><b>then </b>
 <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">h</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="group_1.html#E2:75"><i><font color="Green">E60</font></i></a>, , <a class="txt" href="group_1.html#E5:75_1_1_2_1_1_2"><i><font color="Green">E56</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:75_1_1_2_1_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">h</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E6:75_1_1_2_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><i><font color="Green">E58</font></i>: <font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E6:75_1_1_2_1_1_2"><i><font color="Green">E57</font></i></a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font>
<b>by </b><i/>
;<br/><a NAME="E8:75_1_1_2_1_1_2"/>
<font color="Maroon">m</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#T41">NAT_1:41</a></i>;<br/><a NAME="E9:75_1_1_2_1_1_2"/><b>then </b>
 <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">h</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="group_1.html#T7" target="_self">Th7</a>, <a class="txt" href="group_1.html#E11:75_1_1_2_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="group_1.html#E2:75"><i><font color="Green">E60</font></i></a>, <a class="ref" href="group_1.html#D5" target="_self">Def5</a></i>;<br/><b>then consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:75_1_1_2_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">h</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E12:75_1_1_2_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">h</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E13:75_1_1_2_1_1_2"/>
 <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">h</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/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:75_1_1_2"/>
 <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">h</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:75"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4">NAT_1:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:75"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:75"/>
( <font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">G</font> implies  <a href="group_1.html#K7" target="_self">ord</a> <font color="Maroon">h</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 ;<br/>


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