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<b>let </b><font color="Maroon">G</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<a NAME="E1:41"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" target="_self">.:</a> <font color="Maroon">G</font>,<font color="Maroon">A</font>) <a href="funct_2.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">G</font>) <a href="monoid_1.html#K8" target="_self">.:</a> <font color="Maroon">A</font> &amp; <font color="Maroon">H<sub>3</sub></font>( <a href="monoid_1.html#K9" target="_self">.:</a> <font color="Maroon">G</font>,<font color="Maroon">A</font>) <a href="hidden.html#R1">=</a>  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">G</font>) )
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:41"/>
( <font color="Maroon">G</font> is <a href="group_1.html#V7">commutative</a> implies  <a href="monoid_1.html#K9" target="_self">.:</a> <font color="Maroon">G</font>,<font color="Maroon">A</font> is <a href="group_1.html#V7">commutative</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G</font>) is <a href="binop_1.html#V1">commutative</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D11">MONOID_0:def 11</a><br/>

<b>hence </b><a NAME="E2:41_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" target="_self">.:</a> <font color="Maroon">G</font>,<font color="Maroon">A</font>) is <a href="binop_1.html#V1">commutative</a>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D11">MONOID_0:def 11</a><br/>


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

<b>thus </b><a NAME="E3:41"/>
( <font color="Maroon">G</font> is <a href="group_1.html#V4">associative</a> implies  <a href="monoid_1.html#K9" target="_self">.:</a> <font color="Maroon">G</font>,<font color="Maroon">A</font> is <a href="group_1.html#V4">associative</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G</font>) is <a href="binop_1.html#V2">associative</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D12">MONOID_0:def 12</a><br/>

<b>hence </b><a NAME="E2:41_2"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" target="_self">.:</a> <font color="Maroon">G</font>,<font color="Maroon">A</font>) is <a href="binop_1.html#V2">associative</a>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D12">MONOID_0:def 12</a><br/>


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

<b>thus </b><a NAME="E4:41"/>
( <font color="Maroon">G</font> is <a href="monoid_0.html#V10">idempotent</a> implies  <a href="monoid_1.html#K9" target="_self">.:</a> <font color="Maroon">G</font>,<font color="Maroon">A</font> is <a href="monoid_0.html#V10">idempotent</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G</font>) is <a href="binop_1.html#V3">idempotent</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D13">MONOID_0:def 13</a><br/>

<b>hence </b><a NAME="E2:41_3"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" target="_self">.:</a> <font color="Maroon">G</font>,<font color="Maroon">A</font>) is <a href="binop_1.html#V3">idempotent</a>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D13">MONOID_0:def 13</a><br/>


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

<b>thus </b><a NAME="E5:41"/>
( <font color="Maroon">G</font> is <a href="monoid_0.html#V13">invertible</a> implies  <a href="monoid_1.html#K9" target="_self">.:</a> <font color="Maroon">G</font>,<font color="Maroon">A</font> is <a href="monoid_0.html#V13">invertible</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_4"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G</font>) is <a href="monoid_0.html#V5">invertible</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D16">MONOID_0:def 16</a><br/>

<b>hence </b><a NAME="E2:41_4"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" target="_self">.:</a> <font color="Maroon">G</font>,<font color="Maroon">A</font>) is <a href="monoid_0.html#V5">invertible</a>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D16">MONOID_0:def 16</a><br/>


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

<b>thus </b><a NAME="E6:41"/>
( <font color="Maroon">G</font> is <a href="monoid_0.html#V16">cancelable</a> implies  <a href="monoid_1.html#K9" target="_self">.:</a> <font color="Maroon">G</font>,<font color="Maroon">A</font> is <a href="monoid_0.html#V16">cancelable</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_5"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G</font>) is <a href="monoid_0.html#V8">cancelable</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D19">MONOID_0:def 19</a><br/>

<b>hence </b><a NAME="E2:41_5"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" target="_self">.:</a> <font color="Maroon">G</font>,<font color="Maroon">A</font>) is <a href="monoid_0.html#V8">cancelable</a>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D19">MONOID_0:def 19</a><br/>


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

<b>assume </b><a NAME="E7:41"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G</font>) is <a href="monoid_0.html#V9">uniquely-decomposable</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D20">MONOID_0:def 20</a><br/>

<b>hence </b><a NAME="E8:41"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" target="_self">.:</a> <font color="Maroon">G</font>,<font color="Maroon">A</font>) is <a href="monoid_0.html#V9">uniquely-decomposable</a>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D20">MONOID_0:def 20</a><br/>


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