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<b>set </b><font color="Maroon" title="c4">E</font> =  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">H</font>) <b>-&gt; </b>  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font>, <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> =  <a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> $1,<font color="Maroon" title="c2">p</font>;<br/>
<a NAME="E1:52_1_1"/><i><font color="Green" title="E38">A1</font></i>: 
for <font color="Olive" title="b1">h1</font>, <font color="Olive" title="b2">h2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">H</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h2</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">h1</font>) <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">h2</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">h1</font>, <font color="Maroon" title="c6">h2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">H</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">h1</font>) <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">h2</font>)</span><br/>

<b>set </b><font color="Maroon" title="c7">f12</font> =  <a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font></span>)</span>,<font color="Maroon" title="c2">p</font>;<br/>
<b>set </b><font color="Maroon" title="c8">f1</font> =  <a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c2">p</font>;<br/>
<b>set </b><font color="Maroon" title="c9">f2</font> =  <a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font>;<br/>
<a NAME="E1:52_1_1_1"/><i><font color="Green" title="E39">A2</font></i>: 
(  <a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font></span>)</span>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font></span>)</span> &amp;  <a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font></span>)</span> &amp;  <a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/>
<a NAME="E2:52_1_1_1"/><b>then </b><i><font color="Green" title="E40">A3</font></i>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E3:52_1_1_1"/><i><font color="Green" title="E41">A4</font></i>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> )
 
<b>by </b><i><a class="txt" href="#E1:52_1_1_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E4:52_1_1_1"/><i><font color="Green" title="E42">A5</font></i>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font></span>)</span>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> )
 
<b>by </b><i><a class="txt" href="#E1:52_1_1_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<div><i><font color="Green" title="E43">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span> implies ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c2">p</font></span>)</span> ) ) &amp; ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c2">p</font></span>)</span> implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span> ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c2">p</font></span>)</span> implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span> )</span>
<div class="add"><b>assume </b><a NAME="E1:52_1_1_1_1_1"/><i><font color="Green" title="E44">A7</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c2">p</font></span>)</span> )</span><br/><b>hence </b><a NAME="E2:52_1_1_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:52_1_1_1"><i><font color="Green" title="E41">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c2">p</font></span>)</span></span><br/><a NAME="E3:52_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:52_1_1_1"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="#E1:52_1_1_1_1_1"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E4:52_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:52_1_1_1"><i><font color="Green" title="E40">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:52_1_1_1_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c2">p</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span></span><br/><b>hence </b><a NAME="E3:52_1_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:52_1_1_1"><i><font color="Green" title="E42">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span> implies <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:52_1_1_1_2"/><i><font color="Green" title="E44">A8</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><br/><b>then reconsider </b><font color="Maroon" title="c11">P1'</font> = <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> ;<br/><b>consider </b><font color="Maroon" title="c12">P2'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c13">H1'</font>, <font color="Maroon" title="c14">H2'</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c15">g2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:52_1_1_1_2"/><i><font color="Green" title="E45">A9</font></i>: 
( <font color="Maroon" title="c12">P2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">P1'</font> &amp; <font color="Maroon" title="c11">P1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">H1'</font> &amp; <font color="Maroon" title="c12">P2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">H2'</font> &amp; <font color="Maroon" title="c6">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g2</font> &amp; <font color="Maroon" title="c14">H2'</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c13">H1'</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c15">g2</font> )
 <b>by </b><i><a class="ref" href="group_10.html#D21" target="_self" title="GROUP_10:def.21">Def21</a></i>;<br/><a NAME="E5:52_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:52_1_1_1"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="#E1:52_1_1_1_2"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">P1''</font> = <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> ;<br/><b>consider </b><font color="Maroon" title="c17">P2''</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c18">H1''</font>, <font color="Maroon" title="c19">H2''</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c20">g1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E8:52_1_1_1_2"/><i><font color="Green" title="E46">A10</font></i>: 
( <font color="Maroon" title="c17">P2''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">P1''</font> &amp; <font color="Maroon" title="c16">P1''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">H1''</font> &amp; <font color="Maroon" title="c17">P2''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">H2''</font> &amp; <font color="Maroon" title="c5">h1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">g1</font> &amp; <font color="Maroon" title="c19">H2''</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c18">H1''</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c20">g1</font> )
 <b>by </b><i><a class="ref" href="group_10.html#D21" target="_self" title="GROUP_10:def.21">Def21</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">P1'''</font> = <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="#E1:52_1_1_1_2"><i><font color="Green" title="E44">A8</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c22">P2'''</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c23">H1'''</font>, <font color="Maroon" title="c24">H2'''</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c25">g3</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E11:52_1_1_1_2"/><i><font color="Green" title="E47">A11</font></i>: 
( <font color="Maroon" title="c22">P2'''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">P1'''</font> &amp; <font color="Maroon" title="c21">P1'''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">H1'''</font> &amp; <font color="Maroon" title="c22">P2'''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">H2'''</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">g3</font> &amp; <font color="Maroon" title="c24">H2'''</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c23">H1'''</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c25">g3</font> )
 <b>by </b><i><a class="ref" href="group_10.html#D21" target="_self" title="GROUP_10:def.21">Def21</a></i>;<br/><a NAME="E12:52_1_1_1_2"/>
<font color="Maroon" title="c25">g3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E11:52_1_1_1_2"><i><font color="Green" title="E47">A11</font></i></a>, <a class="ref" href="group_1.html#T25" title="GROUP_1:th.25">GROUP_1:25</a></i>;<br/><b>then </b><font color="Maroon" title="c22">P2'''</font> = 
<font color="Maroon" title="c23">H1'''</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">g2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">g1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:52_1_1_1_2"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="#E8:52_1_1_1_2"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="#E11:52_1_1_1_2"><i><font color="Green" title="E47">A11</font></i></a>, <a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></i>
<br/>.= 
<font color="Maroon" title="c17">P2''</font>
<b>by </b><i><a class="txt" href="#E4:52_1_1_1_2"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="#E8:52_1_1_1_2"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="#E11:52_1_1_1_2"><i><font color="Green" title="E47">A11</font></i></a>, <a class="ref" href="group_3.html#T72" title="GROUP_3:th.72">GROUP_3:72</a></i>
;<br/><b>hence </b><a NAME="E14:52_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c5">h1</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Maroon" title="c6">h2</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:52_1_1_1_2"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="#E11:52_1_1_1_2"><i><font color="Green" title="E47">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:52_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">h2</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">h1</font>) <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">h2</font>)
 <b>by </b><i><a class="txt" href="#E5:52_1_1_1"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="funct_1.html#T20" title="FUNCT_1:th.20">FUNCT_1:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:52_1_1"/><i><font color="Green" title="E39">A12</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c5">f</font> =  <a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font></span>)</span>,<font color="Maroon" title="c2">p</font>;<br/>
<a NAME="E1:52_1_1_2"/>
 <a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font></span>)</span>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/>
<a NAME="E2:52_1_1_2"/><b>then </b><i><font color="Green" title="E40">A13</font></i>: 
 ex <font color="Olive" title="b1">f'</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font></span>)</span>,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f'</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> implies <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> )</span><br/><b>assume </b><a NAME="E1:52_1_1_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<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"> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font></span><br/><b>then reconsider </b><font color="Maroon" title="c7">P1</font> = <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font> ;<br/><b>consider </b><font color="Maroon" title="c8">P2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c9">H1</font>, <font color="Maroon" title="c10">H2</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c11">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:52_1_1_2_1"/><i><font color="Green" title="E41">A14</font></i>: 
( <font color="Maroon" title="c8">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">P1</font> &amp; <font color="Maroon" title="c7">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">H1</font> &amp; <font color="Maroon" title="c8">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H2</font> &amp; <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> &amp; <font color="Maroon" title="c10">H2</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c9">H1</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c11">g</font> )
 <b>by </b><i><a class="ref" href="group_10.html#D21" target="_self" title="GROUP_10:def.21">Def21</a></i>;<br/><a NAME="E5:52_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <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="c3">H</font>
 <b>by </b><i><a class="ref" href="group_1.html#T16" title="GROUP_1:th.16">GROUP_1:16</a></i>;<br/><a NAME="E6:52_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c11">g</font> <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="#E4:52_1_1_2_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="ref" href="group_2.html#T53" title="GROUP_2:th.53">GROUP_2:53</a></i>;<br/><b>hence </b><a NAME="E7:52_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font></span>)</span>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="#E4:52_1_1_2_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="ref" href="group_3.html#T73" title="GROUP_3:th.73">GROUP_3:73</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:52_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:52_1_1_2"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:52_1_1"/>
 ex <font color="Olive" title="b1">T</font> being  <a href="group_10.html#NM1" title="GROUP_10:NM.1">LeftOperation</a> of <font color="Maroon" title="c3">H</font>,<span class="p1">(<span class="default"><a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font></span>)</span> st <br/>for <font color="Olive" title="b2">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">H</font> holds  <font color="Olive" title="b1">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">h</font>)
 
<b>from </b><i><a class="ref" href="group_10.html#S1" target="_self" title="GROUP_10:sch.1">GROUP_10:sch 1</a>(<a class="txt" href="#E2:52_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="#E1:52_1_1"><i><font color="Green" title="E38">A1</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:52_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="group_10.html#NM1" title="GROUP_10:NM.1">LeftOperation</a> of <font color="Maroon" title="c3">H</font>,<span class="p1">(<span class="default"><a href="group_10.html#K15" title="GROUP_10:func.15">the_sylow_p-subgroups_of_prime</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c1">G</font></span>)</span> st <br/>for <font color="Olive" title="b2">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">H</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K16" title="GROUP_10:func.16">the_left_translation_of</a> <font color="Olive" title="b2">h</font>,<font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
