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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">H</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">H</font><br/> for <font color="Olive" title="b3">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> &amp; <font color="Maroon" title="c1">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> holds <br/> <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b3">b</font></span><br/><b>let </b><font color="Maroon" title="c2">H</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font><br/> for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> holds <br/> <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b2">b</font></span><br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  st <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Maroon" title="c1">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> holds <br/> <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Olive" title="b1">b</font></span><br/><b>let </b><font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c1">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> implies  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c4">b</font> )</span><br/>







<b>assume </b><a NAME="E1:3"/><i><font color="Green" title="E1">A1</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c1">G</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c4">b</font></span><br/>

<a NAME="E2:3"/>
<font color="Maroon" title="c2">H</font> is <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>
 
<b>by </b><i><a class="txt" href="group_8.html#E1:3"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="group_2.html#T48" title="GROUP_2:th.48">GROUP_2:48</a></i>;<br/>
<a NAME="E3:3"/><b>then </b><i><font color="Green" title="E2">A2</font></i>: 
( not <font color="Maroon" title="c3">a</font> <a href="group_1.html#NR2" title="GROUP_1:NR.2">is_of_order_0</a>  &amp; not <font color="Maroon" title="c4">b</font> <a href="group_1.html#NR2" title="GROUP_1:NR.2">is_of_order_0</a>  )
 
<b>by </b><i><a class="txt" href="group_8.html#E1:3"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="gr_cy_1.html#T26" title="GR_CY_1:th.26">GR_CY_1:26</a></i>;<br/>
<a NAME="E4:3"/><b>then </b><i><font color="Green" title="E3">A3</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">a</font></span>)</span> <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="c2">H</font>
 
<b>by </b><i><a class="ref" href="group_1.html#D12" title="GROUP_1:def.12">GROUP_1:def 12</a></i>;<br/>
<a NAME="E5:3"/><i><font color="Green" title="E4">A4</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c4">b</font></span>)</span> <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>
 
<b>by </b><i><a class="txt" href="group_8.html#E3:3"><i><font color="Green" title="E2">A2</font></i></a>, <a class="ref" href="group_1.html#D12" title="GROUP_1:def.12">GROUP_1:def 12</a></i>;<br/>
<a NAME="E6:3"/>
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_8.html#E1:3"><i><font color="Green" title="E1">A1</font></i></a>, <a class="ref" href="group_4.html#T3" title="GROUP_4:th.3">GROUP_4:3</a></i>;<br/>
<a NAME="E7:3"/><b>then </b><i><font color="Green" title="E5">A5</font></i>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c4">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="group_8.html#E4:3"><i><font color="Green" title="E3">A3</font></i></a>, <a class="txt" href="group_8.html#E5:3"><i><font color="Green" title="E4">A4</font></i></a>, <a class="ref" href="group_1.html#T86" title="GROUP_1:th.86">GROUP_1:86</a>, <a class="ref" href="group_2.html#T53" title="GROUP_2:th.53">GROUP_2:53</a></i>;<br/>
<a NAME="E8:3"/>
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c4">b</font></span>)</span> <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>
 
<b>by </b><i><a class="txt" href="group_8.html#E1:3"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="group_8.html#E5:3"><i><font color="Green" title="E4">A4</font></i></a>, <a class="ref" href="group_4.html#T3" title="GROUP_4:th.3">GROUP_4:3</a></i>;<br/>
<a NAME="E9:3"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c4">b</font></span>)</span> <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="c2">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T53" title="GROUP_2:th.53">GROUP_2:53</a></i>;<br/>
<a NAME="E10:3"/><b>then </b>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">a</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c4">b</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T86" title="GROUP_1:th.86">GROUP_1:86</a></i>;<br/>
<b>hence </b><a NAME="E11:3"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="group_8.html#E7:3"><i><font color="Green" title="E5">A5</font></i></a>, <a class="ref" href="nat_d.html#T5" title="NAT_D:th.5">NAT_D:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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