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<b>let </b><font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c2">H</font> be  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c3">p</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c4">I</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>let </b><font color="Maroon" title="c5">P</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">H</font>;<br/>







<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 ;<br/>

<b>assume </b><a NAME="E2:53"/>
<font color="Maroon" title="c4">I</font> <a href="group_10.html#R5" title="GROUP_10:pred.5">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c3">p</font>
 ;<br/>

<a NAME="E3:53"/><b>then </b><i><font color="Green" title="E41">A2</font></i>: 
( <font color="Maroon" title="c4">I</font> <a href="group_10.html#R4" title="GROUP_10:pred.4">is_p-group_of_prime</a> <font color="Maroon" title="c3">p</font> &amp; not <font color="Maroon" title="c3">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">I</font> )
 
<b>by </b><i><a class="ref" href="group_10.html#D19" target="_self" title="GROUP_10:def.19">Def19</a></i>;<br/>
<b>assume </b><a NAME="E4:53"/>
<font color="Maroon" title="c2">H</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">H'</font> = <font color="Maroon" title="c2">H</font> as    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> ;<br/>
<a NAME="E6:53"/>
 ex <font color="Olive" title="b1">I'</font> being <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> st <br/>( <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">I'</font> &amp; <font color="Olive" title="b1">I'</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_p-group_of_prime</a> <font color="Maroon" title="c3">p</font> )
 
<b>by </b><i><a class="txt" href="group_10.html#E3:53"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def18</a></i>;<br/>
<a NAME="E7:53"/><b>then </b><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="group_10.html#R4" title="GROUP_10:pred.4">is_p-group_of_prime</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E1:53"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def18</a></i>;<br/>
<a NAME="E8:53"/>
 <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c6">H'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_10.html#E1:53"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="group_2.html#T179" title="GROUP_2:th.179">GROUP_2:179</a></i>;<br/>
<a NAME="E9:53"/><b>then </b>
not <font color="Maroon" title="c3">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="group_2.html#K17" title="GROUP_2:func.17">index</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E3:53"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></i>;<br/>
<b>hence </b><a NAME="E10:53"/>
<font color="Maroon" title="c5">P</font> <a href="group_10.html#R5" title="GROUP_10:pred.5">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c3">p</font>
 <b>by </b><i><a class="txt" href="group_10.html#E7:53"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="group_10.html#D19" target="_self" title="GROUP_10:def.19">Def19</a></i>;<br/>


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