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<div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/>

<b>thus </b><a NAME="E1:2"/>
( <font color="Maroon">G</font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a> implies the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> is <a href="binop_1.html#V1">commutative</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">G</font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a>
 ;<br/>

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



<b>thus </b>the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>

<br/>.= 
<font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
<b>by </b><i>, </i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>

;<br/>


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

<b>assume </b><a NAME="E2:2"/><i><font color="Green">E39</font></i>: 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">G</font> is <a href="binop_1.html#V1">commutative</a>
 ;<br/>

<a NAME="E3:2"/>
<font color="Maroon">G</font> is <a href="group_1.html#V7">commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

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



<b>thus </b><a NAME="E1:2_2"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="binop_1.html#D2">BINOP_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:2"/>
<font color="Maroon">G</font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a>
 ;<br/>


</div>
