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<b>thus </b><a NAME="E1:14_1_1"/>
( <font color="Maroon" title="c1">G</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a> implies the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">G</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_1_1"/><i><font color="Green" title="E8">A1</font></i>: 
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="group_1.html#K1" title="GROUP_1:func.1">*</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="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a><br/>

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13" title="BINOP_1:def.13">BINOP_1:def 13</a><br/><b>let </b><font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>



<a NAME="E2:14_1_1_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">a</font> )
 
;<br/>
<b>hence </b><a NAME="E3:14_1_1_1"/>
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:14_1_1_1"><i><font color="Green" title="E8">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:14_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
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="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13" title="BINOP_1:def.13">BINOP_1:def 13</a><br/>

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a><br/><b>let </b><font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>thus </b><a NAME="E3:14_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:14_1_1"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>


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