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<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="c2">H</font>) <b>-&gt; </b>  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font>, <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font> =  <a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> $1,<font color="Maroon" title="c3">P</font>;<br/>
<a NAME="E1:19_1_1"/><i><font color="Green" title="E11">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="c2">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="relset_1.html#R2" title="RELSET_1: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="19_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">h1</font>, <font color="Maroon" title="c5">h2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font>;<br/>

<b>set </b><font color="Maroon" title="c6">f12</font> =  <a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h2</font></span>)</span>,<font color="Maroon" title="c3">P</font>;<br/>
<b>set </b><font color="Maroon" title="c7">f1</font> =  <a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c4">h1</font>,<font color="Maroon" title="c3">P</font>;<br/>
<b>set </b><font color="Maroon" title="c8">f2</font> =  <a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c5">h2</font>,<font color="Maroon" title="c3">P</font>;<br/>
<a NAME="E1:19_1_1_1"/><i><font color="Green" title="E12">A2</font></i>: 
(  <a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h2</font></span>)</span>,<font color="Maroon" title="c3">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_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font></span>)</span> &amp;  <a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c4">h1</font>,<font color="Maroon" title="c3">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_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font></span>)</span> &amp;  <a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c5">h2</font>,<font color="Maroon" title="c3">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_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</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="E2:19_1_1_1"/><b>then </b><i><font color="Green" title="E13">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#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c4">h1</font>,<font color="Maroon" title="c3">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_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</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_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</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="E3:19_1_1_1"/><i><font color="Green" title="E14">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#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c5">h2</font>,<font color="Maroon" title="c3">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_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</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_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="txt" href="group_10.html#E1:19_1_1_1"><i><font color="Green" title="E12">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="E4:19_1_1_1"/><i><font color="Green" title="E15">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#K8" title="GROUP_10:func.8">the_left_translation_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h2</font></span>)</span>,<font color="Maroon" title="c3">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_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</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_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="txt" href="group_10.html#E1:19_1_1_1"><i><font color="Green" title="E12">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="E16">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:19_1_1_1_1_1"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c9">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#K8" title="GROUP_10:func.8">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h2</font></span>)</span>,<font color="Maroon" title="c3">P</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E2:19_1_1_1_1_1"/>
<font color="Maroon" title="c9">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#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c5">h2</font>,<font color="Maroon" title="c3">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E3:19_1_1_1"><i><font color="Green" title="E14">A4</font></i></a></i>;<br/><a NAME="E3:19_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c5">h2</font>,<font color="Maroon" title="c3">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c5">h2</font>,<font color="Maroon" title="c3">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E3:19_1_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="group_10.html#E1:19_1_1_1_1_1"><i><font color="Green" title="E17">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="E4:19_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c5">h2</font>,<font color="Maroon" title="c3">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c4">h1</font>,<font color="Maroon" title="c3">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E2:19_1_1_1"><i><font color="Green" title="E13">A3</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:19_1_1_1_1"/>
( <font color="Maroon" title="c9">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#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c5">h2</font>,<font color="Maroon" title="c3">P</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c5">h2</font>,<font color="Maroon" title="c3">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c4">h1</font>,<font color="Maroon" title="c3">P</font></span>)</span> )
 ;<br/><b>hence </b><a NAME="E3:19_1_1_1_1"/>
<font color="Maroon" title="c9">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#K8" title="GROUP_10:func.8">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h2</font></span>)</span>,<font color="Maroon" title="c3">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E4:19_1_1_1"><i><font color="Green" title="E15">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:19_1_1_1_2"/><i><font color="Green" title="E17">A8</font></i>: 
<font color="Maroon" title="c9">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#K8" title="GROUP_10:func.8">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h2</font></span>)</span>,<font color="Maroon" title="c3">P</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">P1'</font> = <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font> ;<br/><b>consider </b><font color="Maroon" title="c11">P2'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c12">A1'</font>, <font color="Maroon" title="c13">A2'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c14">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="E4:19_1_1_1_2"/><i><font color="Green" title="E18">A9</font></i>: 
( <font color="Maroon" title="c11">P2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c5">h2</font>,<font color="Maroon" title="c3">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">P1'</font> &amp; <font color="Maroon" title="c13">A2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g'</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <font color="Maroon" title="c12">A1'</font> &amp; <font color="Maroon" title="c12">A1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P1'</font> &amp; <font color="Maroon" title="c13">A2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">P2'</font> &amp; <font color="Maroon" title="c14">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h2</font> )
 <b>by </b><i><a class="ref" href="group_10.html#D8" target="_self" title="GROUP_10:def.8">Def8</a></i>;<br/><a NAME="E5:19_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c5">h2</font>,<font color="Maroon" title="c3">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c5">h2</font>,<font color="Maroon" title="c3">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E3:19_1_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="group_10.html#E1:19_1_1_1_2"><i><font color="Green" title="E17">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="c15">P1''</font> = <span class="p1">(<span class="default"><a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c5">h2</font>,<font color="Maroon" title="c3">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font> ;<br/><b>consider </b><font color="Maroon" title="c16">P2''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c17">A1''</font>, <font color="Maroon" title="c18">A2''</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c19">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="E8:19_1_1_1_2"/><i><font color="Green" title="E19">A10</font></i>: 
( <font color="Maroon" title="c16">P2''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c4">h1</font>,<font color="Maroon" title="c3">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">P1''</font> &amp; <font color="Maroon" title="c18">A2''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">g''</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <font color="Maroon" title="c17">A1''</font> &amp; <font color="Maroon" title="c17">A1''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">P1''</font> &amp; <font color="Maroon" title="c18">A2''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">P2''</font> &amp; <font color="Maroon" title="c19">g''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h1</font> )
 <b>by </b><i><a class="ref" href="group_10.html#D8" target="_self" title="GROUP_10:def.8">Def8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">P1'''</font> = <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font> <b>by </b><i><a class="txt" href="group_10.html#E1:19_1_1_1_2"><i><font color="Green" title="E17">A8</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c21">P2'''</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c22">A1'''</font>, <font color="Maroon" title="c23">A2'''</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c24">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="E11:19_1_1_1_2"/><i><font color="Green" title="E20">A11</font></i>: 
( <font color="Maroon" title="c21">P2'''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h2</font></span>)</span>,<font color="Maroon" title="c3">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">P1'''</font> &amp; <font color="Maroon" title="c23">A2'''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">g'''</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <font color="Maroon" title="c22">A1'''</font> &amp; <font color="Maroon" title="c22">A1'''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">P1'''</font> &amp; <font color="Maroon" title="c23">A2'''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">P2'''</font> &amp; <font color="Maroon" title="c24">g'''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h2</font> )
 <b>by </b><i><a class="ref" href="group_10.html#D8" target="_self" title="GROUP_10:def.8">Def8</a></i>;<br/><a NAME="E12:19_1_1_1_2"/>
<font color="Maroon" title="c24">g'''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">g''</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">g'</font>
 <b>by </b><i><a class="txt" href="group_10.html#E4:19_1_1_1_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="group_10.html#E8:19_1_1_1_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="group_10.html#E11:19_1_1_1_2"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="group_2.html#T52" title="GROUP_2:th.52">GROUP_2:52</a></i>;<br/><b>hence </b><a NAME="E13:19_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h2</font></span>)</span>,<font color="Maroon" title="c3">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_10.html#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c4">h1</font>,<font color="Maroon" title="c3">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#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Maroon" title="c5">h2</font>,<font color="Maroon" title="c3">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E4:19_1_1_1_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="group_10.html#E8:19_1_1_1_2"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="group_10.html#E11:19_1_1_1_2"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="group_2.html#T38" title="GROUP_2:th.38">GROUP_2:38</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:19_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">h2</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">h1</font>) <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">h2</font>)
 <b>by </b><i><a class="txt" href="group_10.html#E5:19_1_1_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="funct_1.html#T20" title="FUNCT_1:th.20">FUNCT_1:20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:19_1_1"/><i><font color="Green" title="E12">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="c2">H</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c4">f</font> =  <a href="group_10.html#K8" title="GROUP_10:func.8">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="c2">H</font></span>)</span>,<font color="Maroon" title="c3">P</font>;<br/>
<a NAME="E1:19_1_1_2"/>
 <a href="group_10.html#K8" title="GROUP_10:func.8">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="c2">H</font></span>)</span>,<font color="Maroon" title="c3">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_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</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="E2:19_1_1_2"/><b>then </b><i><font color="Green" title="E13">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#K8" title="GROUP_10:func.8">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="c2">H</font></span>)</span>,<font color="Maroon" title="c3">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_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</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_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</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="19_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:19_1_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">P1</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font> ;<br/><b>consider </b><font color="Maroon" title="c7">P2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c8">A1</font>, <font color="Maroon" title="c9">A2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c10">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="E4:19_1_1_2_1"/><i><font color="Green" title="E14">A14</font></i>: 
( <font color="Maroon" title="c7">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_10.html#K8" title="GROUP_10:func.8">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="c2">H</font></span>)</span>,<font color="Maroon" title="c3">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">P1</font> &amp; <font color="Maroon" title="c9">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">g</font> <a href="group_2.html#K3" title="GROUP_2:func.3">*</a> <font color="Maroon" title="c8">A1</font> &amp; <font color="Maroon" title="c8">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P1</font> &amp; <font color="Maroon" title="c9">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">P2</font> &amp; <font color="Maroon" title="c10">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="c2">H</font> )
 <b>by </b><i><a class="ref" href="group_10.html#D8" target="_self" title="GROUP_10:def.8">Def8</a></i>;<br/><a NAME="E5:19_1_1_2_1"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">H</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="ref" href="group_2.html#T53" title="GROUP_2:th.53">GROUP_2:53</a></i>;<br/><b>hence </b><a NAME="E6:19_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="group_10.html#K8" title="GROUP_10:func.8">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="c2">H</font></span>)</span>,<font color="Maroon" title="c3">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="group_10.html#E4:19_1_1_2_1"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="group_2.html#T43" title="GROUP_2:th.43">GROUP_2:43</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:19_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="c2">H</font>) <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_10.html#E2:19_1_1_2"><i><font color="Green" title="E13">A13</font></i></a>, <a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:19_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="c2">H</font>,<span class="p1">(<span class="default"><a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</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="c2">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="group_10.html#E2:19_1_1"><i><font color="Green" title="E12">A12</font></i></a>, <a class="txt" href="group_10.html#E1:19_1_1"><i><font color="Green" title="E11">A1</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:19_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="c2">H</font>,<span class="p1">(<span class="default"><a href="group_2.html#K14" title="GROUP_2:func.14">Left_Cosets</a> <font color="Maroon" title="c3">P</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="c2">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#K8" title="GROUP_10:func.8">the_left_translation_of</a> <font color="Olive" title="b2">h</font>,<font color="Maroon" title="c3">P</font>
 ;<br/>


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