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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">G1</font> be    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<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>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">p</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>assume </b><b>that </b><br/><a NAME="E1:11"/><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="E2:11"/><i><font color="Green">E49</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>
 <b>and </b><br/><a NAME="E3:11"/><i><font color="Green">E50</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="E4:11"/><i><font color="Green">E54</font></i>: 
 <a href="group_1.html#K9">ord</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E5:11"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">G1</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>
 ;<br/>

<b>set </b><font color="Maroon">z</font> = <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font>;<br/>
<a NAME="E6:11"/>
<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/>
<b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">l</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:11"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">l</font>
 <b>and </b><br/><a NAME="E9:11"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T42">NAT_1:42</a></i>;<br/>
<a NAME="E10:11"/>
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:11_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">G1</font> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T48">GROUP_2:48</a></i>;<br/>
<a NAME="E3:11_1"/>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</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/>;<br/>
<b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font> <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</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="nat_1.html#K2">*</a> <font color="Maroon">p</font></span>)</span> = 
<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>
<b>by </b><i><a class="ref" href="group_1.html#T50">GROUP_1:50</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="group_4.html#T3">GROUP_4:3</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G1</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_2.html#T53">GROUP_2:53</a></i>
;<br/>
<b>then </b><i><font color="Green">E68</font></i>:  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</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"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</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> <font color="Maroon">l</font></span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#T48">GROUP_1:48</a></i>
<br/>.= 
<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> <font color="Maroon">m</font></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> <font color="Maroon">l</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T50">GROUP_1:50</a></i>
<br/>.= 
<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> <font color="Maroon">m</font></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> <font color="Maroon">l</font></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"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></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> <font color="Maroon">l</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T42">GROUP_1:42</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">l</font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<a NAME="E7:11_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">a</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="E8:11_1"/>
 <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , , <a class="ref" href="gr_cy_1.html#T27">GR_CY_1:27</a></i>;<br/>
<b>hence </b><a NAME="E9:11_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="group_1.html#D12">GROUP_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:11"/>
<font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</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/>


</div>
