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<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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">A</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  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">A</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#V10" title="MONOID_0:attr.10">idempotent</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">A</font>) is  <a href="monoid_0.html#V10" title="MONOID_0:attr.10">idempotent</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">A</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#V16" title="MONOID_0:attr.16">cancelable</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">A</font>) is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</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  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">A</font>) is  <a href="monoid_0.html#V17" title="MONOID_0:attr.17">uniquely-decomposable</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</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  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</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#V10" title="MONOID_0:attr.10">idempotent</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V10" title="MONOID_0:attr.10">idempotent</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</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#V16" title="MONOID_0:attr.16">cancelable</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</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  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V17" title="MONOID_0:attr.17">uniquely-decomposable</a>  ) )</span><br/>



<a NAME="E1:40"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
( <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>)) <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> (<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>)) <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon">H<sub>3</sub></font>( <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>)) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c2">A</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>)) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_1.html#T17" target="_self" title="MONOID_1:th.17">Th18</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:40"/>
( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</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="firebrick">::  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  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</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#V10" title="MONOID_0:attr.10">idempotent</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V10" title="MONOID_0:attr.10">idempotent</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</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#V16" title="MONOID_0:attr.16">cancelable</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</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  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V17" title="MONOID_0:attr.17">uniquely-decomposable</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:40_1"/>
<font color="Maroon">H<sub>1</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="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D11" title="MONOID_0:def.11">MONOID_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:40_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</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="#E1:40"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="monoid_1.html#T7" target="_self" title="MONOID_1:th.7">Th8</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D11" title="MONOID_0:def.11">MONOID_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E3:40"/>
( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V10" title="MONOID_0:attr.10">idempotent</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V10" title="MONOID_0:attr.10">idempotent</a>  ) &amp; ( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</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#V16" title="MONOID_0:attr.16">cancelable</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</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  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V17" title="MONOID_0:attr.17">uniquely-decomposable</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:40_2"/>
<font color="Maroon">H<sub>1</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="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D12" title="MONOID_0:def.12">MONOID_0:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:40_2"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</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="#E1:40"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="monoid_1.html#T8" target="_self" title="MONOID_1:th.8">Th9</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D12" title="MONOID_0:def.12">MONOID_0:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E4:40"/>
( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V10" title="MONOID_0:attr.10">idempotent</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V10" title="MONOID_0:attr.10">idempotent</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</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#V16" title="MONOID_0:attr.16">cancelable</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</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  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V17" title="MONOID_0:attr.17">uniquely-decomposable</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:40_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>) is  <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D13" title="MONOID_0:def.13">MONOID_0:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V10" title="MONOID_0:attr.10">idempotent</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:40_3"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>)) is  <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="monoid_1.html#T11" target="_self" title="MONOID_1:th.11">Th12</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D13" title="MONOID_0:def.13">MONOID_0:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E5:40"/>
( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</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="firebrick">::  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  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</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  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V17" title="MONOID_0:attr.17">uniquely-decomposable</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:40_4"/>
<font color="Maroon">H<sub>1</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="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D16" title="MONOID_0:def.16">MONOID_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:40_4"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</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="#E1:40"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="monoid_1.html#T12" target="_self" title="MONOID_1:th.12">Th13</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D16" title="MONOID_0:def.16">MONOID_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E6:40"/>
( <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  implies  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V17" title="MONOID_0:attr.17">uniquely-decomposable</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:40_5"/>
<font color="Maroon">H<sub>1</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="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D19" title="MONOID_0:def.19">MONOID_0:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font>) is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:40_5"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</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="#E1:40"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="monoid_1.html#T13" target="_self" title="MONOID_1:th.13">Th14</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D19" title="MONOID_0:def.19">MONOID_0:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E7:40"/>
<font color="Maroon">H<sub>1</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="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D20" title="MONOID_0:def.20">MONOID_0:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</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="E8:40"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</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="#E1:40"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="monoid_1.html#T14" target="_self" title="MONOID_1:th.14">Th15</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="monoid_0.html#D20" title="MONOID_0:def.20">MONOID_0:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
