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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/>

<b>thus </b><a NAME="E1:22"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="monoid_0.html#V10" target="_self">idempotent</a> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D15">BINOP_1:def 15</a>,<a class="ref" href="monoid_0.html#D13" target="_self">MONOID_0:def 13</a><br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>

<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E1:22_1"><i><font color="Green">E16</font></i></a></i>
;<br/>


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

<b>assume </b><a NAME="E2:22"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D15">BINOP_1:def 15</a>,<a class="ref" href="monoid_0.html#D13" target="_self">MONOID_0:def 13</a><br/>

<b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font>

<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E2:22"><i><font color="Green">E16</font></i></a></i>
;<br/>


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