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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>;<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>;<br/>



<b>assume </b><a NAME="E1:71"/><i><font color="Green" title="E50">A1</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</a>
 ;<br/>

<a NAME="E2:71"/><b>then </b>
<font color="Maroon" title="c2">H</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">finite</a>
 
<b>by </b><i><a class="ref" href="group_2.html#T48" target="_self" title="GROUP_2:th.48">Th48</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">C</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:71"/><i><font color="Green" title="E51">A2</font></i>: 
( <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H</font> &amp;  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c3">C</font> )
 <b>by </b><i><a class="ref" href="group_1.html#D15" title="GROUP_1:def.15">GROUP_1:def 15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">B</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:71"/><i><font color="Green" title="E52">A3</font></i>: 
( <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> &amp;  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">B</font> )
 <b>by </b><i><a class="txt" href="group_2.html#E1:71"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="group_1.html#D15" title="GROUP_1:def.15">GROUP_1:def 15</a></i>;<br/>
<a NAME="E7:71"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D5" target="_self" title="GROUP_2:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E8:71"/>
 <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c2">H</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="group_1.html#K9" title="GROUP_1:func.9">ord</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="group_2.html#E4:71"><i><font color="Green" title="E51">A2</font></i></a>, <a class="txt" href="group_2.html#E6:71"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="card_1.html#T80" title="CARD_1:th.80">CARD_1:80</a></i>;<br/>


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