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<b>let </b><font color="Maroon">H</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/>

<b>assume </b><a NAME="E1:66_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">H</font> is <a href="realset2.html#V3">trivial</a>
 ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3">GROUP_1:def 3</a>
<div class="add"><b>consider </b><font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font>;<br/><b>take </b><font color="Maroon">e</font> = <font color="Maroon">e</font>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font>;<br/><b>thus </b><a NAME="E2:66_1_1_1"/>
( <font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp; <font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> )
 <b>by </b><i>, <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/><b>take </b><font color="Maroon">g</font> = <font color="Maroon">e</font>;<br/><b>thus </b><a NAME="E3:66_1_1_1"/>
( <font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>by </b><i>, <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E3:66_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> holds  <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="group_1.html#K1">*</a> <font color="Olive">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a><br/>

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

<b>thus </b><a NAME="E4:66_1_1"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>


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