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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <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/>  for <font color="Olive" title="b2">I</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="b3">P</font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">H</font>  st <font color="Olive" title="b2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">P</font> &amp; <font color="Olive" title="b2">I</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Olive" title="b1">p</font> &amp; <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> holds <br/><font color="Olive" title="b3">P</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Olive" title="b1">p</font></span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">I</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">P</font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">H</font>  st <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P</font> &amp; <font color="Olive" title="b1">I</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c3">p</font> &amp; <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> holds <br/><font color="Olive" title="b2">P</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c3">p</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">H</font>  st <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> &amp; <font color="Maroon" title="c4">I</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c3">p</font> &amp; <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> holds <br/><font color="Olive" title="b1">P</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c3">p</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c4">I</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c3">p</font> &amp; <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> implies <font color="Maroon" title="c5">P</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c3">p</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:54"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<font color="Maroon" title="c4">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">I</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c3">p</font> or  not <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> or <font color="Maroon" title="c5">P</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c3">p</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:54"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<font color="Maroon" title="c4">I</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c3">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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> or <font color="Maroon" title="c5">P</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c3">p</font> )</span><br/>

<a NAME="E3:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A3</font></span>: 
<font color="Maroon" title="c4">I</font> is <font color="Maroon" title="c3">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def19</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:54"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">P</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c3">p</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">H9</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="E6:54"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
 <a href="group_2.html#K18" title="GROUP_2:func.18">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#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="group_2.html#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c6">H9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="group_2.html#T149" title="GROUP_2:th.149">GROUP_2:149</a></span>;<br/>
<a NAME="E7:54"/>
 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#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c4">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def19</a></span>;<br/>
<a NAME="E8:54"/><span class="kw">then </span>
 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#K18" title="GROUP_2:func.18">index</a> <font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="nat_d.html#T9" title="NAT_D:th.9">NAT_D:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:54"/>
<font color="Maroon" title="c5">P</font> <a href="group_10.html#R3" title="GROUP_10:pred.3">is_Sylow_p-subgroup_of_prime</a> <font color="Maroon" title="c3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="group_10.html#D18" target="_self" title="GROUP_10:def.18">Def19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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