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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/>

<a NAME="E1:89"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K22" target="_self">bool</a> <font color="Maroon">c<sub>1</sub></font>) <a href="funct_2.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="monoid_1.html#K21" target="_self">.:^2</a>  &amp; <font color="Maroon">H<sub>3</sub></font>( <a href="monoid_1.html#K22" target="_self">bool</a> <font color="Maroon">c<sub>1</sub></font>) <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) )
 
<b>by </b><i><a class="ref" href="monoid_1.html#T56" target="_self">Th56</a></i>;<br/>
<b>thus </b><a NAME="E2:89"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V7">commutative</a> implies  <a href="monoid_1.html#K22" target="_self">bool</a> <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V7">commutative</a> )
 <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">c<sub>1</sub></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="E2:89_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K22" target="_self">bool</a> <font color="Maroon">c<sub>1</sub></font>) is <a href="binop_1.html#V1">commutative</a>
 <b>by </b><i><a class="txt" href="monoid_1.html#E1:89"><i><font color="Green">E53</font></i></a>, <a class="ref" href="monoid_1.html#T50" target="_self">Th50</a></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="E3:89"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V4">associative</a> implies  <a href="monoid_1.html#K22" target="_self">bool</a> <font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V4">associative</a> )
 <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">c<sub>1</sub></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="E2:89_2"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K22" target="_self">bool</a> <font color="Maroon">c<sub>1</sub></font>) is <a href="binop_1.html#V2">associative</a>
 <b>by </b><i><a class="txt" href="monoid_1.html#E1:89"><i><font color="Green">E53</font></i></a>, <a class="ref" href="monoid_1.html#T51" target="_self">Th51</a></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>assume </b><a NAME="E4:89"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></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="E5:89"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="monoid_1.html#K22" target="_self">bool</a> <font color="Maroon">c<sub>1</sub></font>) is <a href="monoid_0.html#V9">uniquely-decomposable</a>
 <b>by </b><i><a class="txt" href="monoid_1.html#E1:89"><i><font color="Green">E53</font></i></a>, <a class="ref" href="monoid_1.html#T55" target="_self">Th55</a></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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