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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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">N</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of  <a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Olive" title="b1">N</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a>  holds <br/><font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">N</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of  <a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c2">N</font> is    <a href="group_6.html#M1" title="GROUP_6:mode.1">Subgroup</a> of  <a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">and </span><br/><a NAME="E2:10"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font> is  <a href="gr_cy_1.html#V1" title="GR_CY_1:attr.1">cyclic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">I</font> = <font color="Maroon" title="c1">G</font> <a href="group_6.html#K5" title="GROUP_6:func.5">./.</a> <font color="Maroon" title="c2">N</font> as   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">S</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">I</font><span class="kw"> such that </span><br/><a NAME="E5:10"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
 for <font color="Olive" title="b1">T</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">I</font>  ex <font color="Olive" title="b2">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Olive" title="b1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Olive" title="b2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="gr_cy_1.html#T18" title="GR_CY_1:th.18">GR_CY_1:18</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">a1</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="E7:10"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
( <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a1</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">N</font> &amp; <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">N</font> <a href="group_2.html#K14" title="GROUP_2:func.14">*</a> <font color="Maroon" title="c5">a1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T21" title="GROUP_6:th.21">GROUP_6:21</a></span>;<br/>
<a NAME="E8:10"/>
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font></span><br/>

<a NAME="E1:10_1"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
( <font color="Maroon" title="c6">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">N</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">I</font> &amp; <font color="Maroon" title="c7">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">N</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">I</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T22" title="GROUP_6:th.22">GROUP_6:22</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">r1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:10_1"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c8">r1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/>
<a NAME="E4:10_1"/>
<font color="Maroon" title="c6">a</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c8">r1</font></span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="groupp_1.html#T8" target="_self" title="GROUPP_1:th.8">Th8</a>, <a class="txt" href="#E3:10_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">c1</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="E6:10_1"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c8">r1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">c1</font> &amp; <font color="Maroon" title="c9">c1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">N</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="groupp_1.html#T9" target="_self" title="GROUPP_1:th.9">Th9</a></span>;<br/>
<a NAME="E7:10_1"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
<font color="Maroon" title="c9">c1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E6:10_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="ref" href="group_2.html#T40" title="GROUP_2:th.40">GROUP_2:40</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">r2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:10_1"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c10">r2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E1:10_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a></span>;<br/>
<a NAME="E10:10_1"/>
<font color="Maroon" title="c7">b</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c2">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="groupp_1.html#T8" target="_self" title="GROUPP_1:th.8">Th8</a>, <a class="txt" href="#E9:10_1"><span class="lab"><font color="Green" title="E19">A9</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">c2</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="E12:10_1"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">c2</font> &amp; <font color="Maroon" title="c11">c2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">N</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="groupp_1.html#T9" target="_self" title="GROUPP_1:th.9">Th9</a></span>;<br/>
<a NAME="E13:10_1"/><span class="lab"><font color="Green" title="E21">A11</font></span>: 
<font color="Maroon" title="c11">c2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E12:10_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="ref" href="group_2.html#T40" title="GROUP_2:th.40">GROUP_2:40</a></span>;<br/>
<font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c8">r1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">c1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">c2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E12:10_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c8">r1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">c2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">c1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="group_5.html#T77" title="GROUP_5:th.77">GROUP_5:77</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c8">r1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">c2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">c1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c8">r1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c10">r2</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">c2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">c1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">r1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c10">r2</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">c2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">c1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T33" title="GROUP_1:th.33">GROUP_1:33</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c8">r1</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">c2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">c1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T33" title="GROUP_1:th.33">GROUP_1:33</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c8">r1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">c2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">c1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c8">r1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">c2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">c1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">c2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c8">r1</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">c1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:10_1"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="ref" href="group_5.html#T77" title="GROUP_5:th.77">GROUP_5:77</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">c2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c8">r1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">c1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c7">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E12:10_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:10_1"/>
<font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:10"/>
<font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D12" title="GROUP_1:def.12">GROUP_1:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
