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<b>let </b><font color="Maroon" title="c1">G</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> ;<br/>

<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E23">A1</font></i>: 
( <font color="Maroon" title="c1">G</font> is <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> &amp; <font color="Maroon" title="c1">G</font> is <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a> )
 ;<br/>

<a NAME="E2:33"/><b>then </b>
<font color="Maroon" title="c1">G</font> is <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a>
 
<b>by </b><i><a class="ref" href="monoid_0.html#T17" target="_self" title="MONOID_0:th.17">Th17</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:33"/><i><font color="Green" title="E24">A2</font></i>: 
for <font color="Olive" title="b1">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 <br/> ( <font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Olive" title="b1">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> )
 <b>by </b><i><a class="ref" href="monoid_0.html#T6" target="_self" title="MONOID_0:th.6">Th6</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">a</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</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="E5:33"/>
( <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E4:33"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>

<a NAME="E6:33"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is <a href="monoid_0.html#V5" title="MONOID_0:attr.5">invertible</a>
 
<b>by </b><i><a class="txt" href="monoid_0.html#E1:33"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="monoid_0.html#D16" target="_self" title="MONOID_0:def.16">Def16</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">l</font>, <font color="Maroon" title="c5">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E8:33"/><i><font color="Green" title="E25">A3</font></i>: 
( <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="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</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="c5">r</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> )
 <b>by </b><i><a class="ref" href="monoid_0.html#D5" target="_self" title="MONOID_0:def.5">Def5</a></i>;<br/>
<a NAME="E9:33"/><i><font color="Green" title="E26">A4</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">r</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="txt" href="monoid_0.html#E8:33"><i><font color="Green" title="E25">A3</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">l</font>
;<br/>

<font color="Maroon" title="c5">r</font> = 
<font color="Maroon" title="c5">r</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E4:33"><i><font color="Green" title="E24">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c4">l</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E1:33"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="monoid_0.html#E9:33"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="monoid_0.html#E28"><i><font color="Green" title="E14">Lm2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">l</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E4:33"><i><font color="Green" title="E24">A2</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E11:33"/>
( <font color="Maroon" title="c3">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">l</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="c2">a</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E8:33"><i><font color="Green" title="E25">A3</font></i></a></i>;<br/>


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