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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</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#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</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#V17" title="MONOID_0:attr.17">uniquely-decomposable</a> implies  <a href="monoid_1.html#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</font> is <a href="monoid_0.html#V17" title="MONOID_0:attr.17">uniquely-decomposable</a> ) )</span><br/>

<a NAME="E1:89"/><i><font color="Green" title="E52">A1</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</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>) <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a>  &amp; <font color="Maroon">H<sub>3</sub></font>( <a href="monoid_1.html#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>) )
 
<b>by </b><i><a class="ref" href="monoid_1.html#T56" target="_self" title="MONOID_1:th.56">Th56</a></i>;<br/>
<b>thus </b><a NAME="E2:89"/>
( <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#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</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#V17" title="MONOID_0:attr.17">uniquely-decomposable</a> implies  <a href="monoid_1.html#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</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="89_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:89_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>
 ; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="monoid_1.html#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a></span><br/>

<b>hence </b><a NAME="E2:89_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</font>) is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 <b>by </b><i><a class="txt" href="#E1:89"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="monoid_1.html#T50" target="_self" title="MONOID_1:th.50">Th50</a></i>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:89"/>
( <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#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</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="89_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:89_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>
 ; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="monoid_1.html#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a></span><br/>

<b>hence </b><a NAME="E2:89_2"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</font>) is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 <b>by </b><i><a class="txt" href="#E1:89"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="monoid_1.html#T51" target="_self" title="MONOID_1:th.51">Th51</a></i>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E4:89"/>
<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>
 ; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="monoid_1.html#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</font> is <a href="monoid_0.html#V17" title="MONOID_0:attr.17">uniquely-decomposable</a></span><br/>

<b>hence </b><a NAME="E5:89"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</font>) is <a href="monoid_0.html#V9" title="MONOID_0:attr.9">uniquely-decomposable</a>
 <b>by </b><i><a class="txt" href="#E1:89"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="monoid_1.html#T55" target="_self" title="MONOID_1:th.55">Th55</a></i>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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