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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>let </b><font color="Maroon" title="c2">M</font> be    <a href="monoid_0.html#M1" title="MONOID_0:mode.1">MonoidalExtension</a> of <font color="Maroon" title="c1">G</font>;<br/>



<a NAME="E1:48"/><i><font color="Green" title="E30">A1</font></i>: 
 <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c2">M</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">G</font> #)
 
<b>by </b><i><a class="ref" href="monoid_0.html#D22" target="_self" title="MONOID_0:def.22">Def22</a></i>;<br/>
<b>thus </b><a NAME="E2:48"/>
( <font color="Maroon" title="c1">G</font> is <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> implies <font color="Maroon" title="c2">M</font> is <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D10" target="_self" title="MONOID_0:def.10">MONOID_0:def 10</a><br/>

<b>hence </b><a NAME="E2:48_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">M</font>) <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a> 
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:48"><i><font color="Green" title="E30">A1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D10" target="_self" title="MONOID_0:def.10">MONOID_0:def 10</a><br/>


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

<b>thus </b><a NAME="E3:48"/>
( <font color="Maroon" title="c1">G</font> is <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> implies <font color="Maroon" title="c2">M</font> is <a href="group_1.html#V7" title="GROUP_1:attr.7">commutative</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D11" target="_self" title="MONOID_0:def.11">MONOID_0:def 11</a><br/>

<b>hence </b><a NAME="E2:48_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">M</font>) is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:48"><i><font color="Green" title="E30">A1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D11" target="_self" title="MONOID_0:def.11">MONOID_0:def 11</a><br/>


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

<b>thus </b><a NAME="E4:48"/>
( <font color="Maroon" title="c1">G</font> is <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> implies <font color="Maroon" title="c2">M</font> is <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_3"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D12" target="_self" title="MONOID_0:def.12">MONOID_0:def 12</a><br/>

<b>hence </b><a NAME="E2:48_3"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">M</font>) is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:48"><i><font color="Green" title="E30">A1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D12" target="_self" title="MONOID_0:def.12">MONOID_0:def 12</a><br/>


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

<b>thus </b><a NAME="E5:48"/>
( <font color="Maroon" title="c1">G</font> is <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a> implies <font color="Maroon" title="c2">M</font> is <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_4"/>
<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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D16" target="_self" title="MONOID_0:def.16">MONOID_0:def 16</a><br/>

<b>hence </b><a NAME="E2:48_4"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">M</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#E1:48"><i><font color="Green" title="E30">A1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D16" target="_self" title="MONOID_0:def.16">MONOID_0:def 16</a><br/>


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

<b>thus </b><a NAME="E6:48"/>
( <font color="Maroon" title="c1">G</font> is <a href="monoid_0.html#V17" title="MONOID_0:attr.17">uniquely-decomposable</a> implies <font color="Maroon" title="c2">M</font> is <a href="monoid_0.html#V17" title="MONOID_0:attr.17">uniquely-decomposable</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_5"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is <a href="monoid_0.html#V9" title="MONOID_0:attr.9">uniquely-decomposable</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D20" target="_self" title="MONOID_0:def.20">MONOID_0:def 20</a><br/>

<b>hence </b><a NAME="E2:48_5"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">M</font>) is <a href="monoid_0.html#V9" title="MONOID_0:attr.9">uniquely-decomposable</a>
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:48"><i><font color="Green" title="E30">A1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D20" target="_self" title="MONOID_0:def.20">MONOID_0:def 20</a><br/>


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

<b>assume </b><a NAME="E7:48"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is <a href="monoid_0.html#V8" title="MONOID_0:attr.8">cancelable</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D19" target="_self" title="MONOID_0:def.19">MONOID_0:def 19</a><br/>

<b>hence </b><a NAME="E8:48"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">M</font>) is <a href="monoid_0.html#V8" title="MONOID_0:attr.8">cancelable</a>
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:48"><i><font color="Green" title="E30">A1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D19" target="_self" title="MONOID_0:def.19">MONOID_0:def 19</a><br/>


</div>
