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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M</font> being   <a href="monoid_0.html#M1" title="MONOID_0:mode.1">MonoidalExtension</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  implies <font color="Olive" title="b1">M</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  implies <font color="Olive" title="b1">M</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  implies <font color="Olive" title="b1">M</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  implies <font color="Olive" title="b1">M</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  ) &amp; ( <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="Olive" title="b1">M</font> is  <a href="monoid_0.html#V17" title="MONOID_0:attr.17">uniquely-decomposable</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  implies <font color="Olive" title="b1">M</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  ) )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  implies <font color="Maroon" title="c2">M</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  implies <font color="Maroon" title="c2">M</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  implies <font color="Maroon" title="c2">M</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  ) &amp; ( <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>  ) &amp; ( <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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  implies <font color="Maroon" title="c2">M</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  ) )</span><br/>



<a NAME="E1:47"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">M</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</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="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">G</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D22" target="_self" title="MONOID_0:def.22">Def22</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:47"/>
( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  implies <font color="Maroon" title="c2">M</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  implies <font color="Maroon" title="c2">M</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  implies <font color="Maroon" title="c2">M</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  ) &amp; ( <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>  ) &amp; ( <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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  implies <font color="Maroon" title="c2">M</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:47_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D10" target="_self" title="MONOID_0:def.10">MONOID_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">M</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:47_1"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">M</font>) is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D10" target="_self" title="MONOID_0:def.10">MONOID_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:47"/>
( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  implies <font color="Maroon" title="c2">M</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  implies <font color="Maroon" title="c2">M</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  ) &amp; ( <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>  ) &amp; ( <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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  implies <font color="Maroon" title="c2">M</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:47_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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D11" target="_self" title="MONOID_0:def.11">MONOID_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">M</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:47_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D11" target="_self" title="MONOID_0:def.11">MONOID_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E4:47"/>
( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  implies <font color="Maroon" title="c2">M</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <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>  ) &amp; ( <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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  implies <font color="Maroon" title="c2">M</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:47_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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D12" target="_self" title="MONOID_0:def.12">MONOID_0:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">M</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:47_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D12" target="_self" title="MONOID_0:def.12">MONOID_0:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E5:47"/>
( <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>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <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>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  implies <font color="Maroon" title="c2">M</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:47_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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D16" target="_self" title="MONOID_0:def.16">MONOID_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">M</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:47_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D16" target="_self" title="MONOID_0:def.16">MONOID_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E6:47"/>
( <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>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  implies <font color="Maroon" title="c2">M</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:47_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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D20" target="_self" title="MONOID_0:def.20">MONOID_0:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">M</font> is  <a href="monoid_0.html#V17" title="MONOID_0:attr.17">uniquely-decomposable</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:47_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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D20" target="_self" title="MONOID_0:def.20">MONOID_0:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E7:47"/>
<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> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D19" target="_self" title="MONOID_0:def.19">MONOID_0:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">M</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a> </span><br/>

<span class="kw">hence </span><a NAME="E8:47"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="monoid_0.html#D19" target="_self" title="MONOID_0:def.19">MONOID_0:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
