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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">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">s</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><a NAME="E1:9"/>
<font color="Maroon">s</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:9"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <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="E4:9"/>
 ex <font color="Olive">i</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</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">s</font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<b>take </b>
<font color="Maroon">p'</font>
;<br/>

<b>thus </b><a NAME="E2:9_1"/>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</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">s</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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:9"/>
<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">s</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/>


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