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<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">H</font> be  <a href="group_1.html#V1">strict</a>  <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">H</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>







<b>assume </b><a NAME="E1:5"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a> )
 ;<br/>

<a NAME="E2:5"/>
<font color="Maroon">H</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="E3:5"/><b>then </b><i><font color="Green">E40</font></i>: 
( not <font color="Maroon">a</font> <a href="group_1.html#NR2" target="_self">is_of_order_0</a>  &amp; not <font color="Maroon">b</font> <a href="group_1.html#NR2" target="_self">is_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="E4:5"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_1.html#D12">GROUP_1:def 12</a></i>;<br/>
<a NAME="E5:5"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">b</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="group_1.html#D12">GROUP_1:def 12</a></i>;<br/>
<a NAME="E6:5"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="group_2.html#T53">GROUP_2:53</a></i>;<br/>
<a NAME="E7:5"/>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:5"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">b</font> <a href="nat_1.html#R1">divides</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="group_1.html#T86">GROUP_1:86</a></i>;<br/>
<a NAME="E9:5"/>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E10:5"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T53">GROUP_2:53</a></i>;<br/>
<a NAME="E11:5"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="nat_1.html#R1">divides</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T86">GROUP_1:86</a></i>;<br/>
<b>thus </b><a NAME="E12:5"/>
 <a href="group_1.html#K7">ord</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T52">NAT_1:52</a></i>;<br/>


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