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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b3">N</font> being  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Olive" title="b3">N</font> &amp; <font color="Olive" title="b2">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Olive" title="b3">N</font> holds <br/><font color="Olive" title="b1">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Olive" title="b2">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Olive" title="b3">N</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A1</font>, <font color="Maroon" title="c3">A2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">N</font> being  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  st <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Olive" title="b1">N</font> &amp; <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Olive" title="b1">N</font> holds <br/><font color="Maroon" title="c2">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Olive" title="b1">N</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">N</font> be   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">N</font> implies <font color="Maroon" title="c2">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">N</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c2">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">N</font>
 <span class="kw">and </span><br/><a NAME="E2:27"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<font color="Maroon" title="c3">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">N</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E4:27"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">N</font>
 <span class="kw">and </span><br/><a NAME="E5:27"/>
<font color="Maroon" title="c2">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E7:27"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c3">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">N</font>
 <span class="kw">and </span><br/><a NAME="E8:27"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c3">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">N</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="group_6.html#T15" target="_self" title="GROUP_6:th.15">Th15</a></span>;<br/>
<font color="Maroon" title="c2">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c3">A2</font> = 
<font color="Maroon" title="c5">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">N</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="group_3.html#T11" title="GROUP_3:th.11">GROUP_3:11</a></span>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="group_2.html#K11" title="GROUP_2:func.11">*</a> <font color="Maroon" title="c4">N</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E8:27"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="group_3.html#T15" title="GROUP_3:th.15">GROUP_3:15</a></span>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">N</font> <a href="group_4.html#K7" title="GROUP_4:func.7">*</a> <font color="Maroon" title="c4">N</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_4.html#T60" title="GROUP_4:th.60">GROUP_4:60</a></span>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="group_2.html#K4" title="GROUP_2:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">N</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T91" title="GROUP_2:th.91">GROUP_2:91</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c4">N</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T127" title="GROUP_2:th.127">GROUP_2:127</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:27"/>
<font color="Maroon" title="c2">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c3">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c4">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D15" title="GROUP_2:def.15">GROUP_2:def 15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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