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<b>consider </b><font color="Maroon" title="c1">p</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>, <font color="Maroon" title="c2">o</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>, <font color="Maroon" title="c3">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<b>take </b><font color="Maroon" title="c4">G</font> =  <a href="algstr_0.html#G4" title="ALGSTR_0:aggr.4">multLoopStr</a>(# <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">e</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c5">e</font> = <font color="Maroon" title="c3">e</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">o</font> = <font color="Maroon" title="c2">o</font> as   <a href="monoid_0.html#NM1" title="MONOID_0:NM.1">BinOp</a> of <font color="Maroon" title="c4">G</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font>;<br/><a NAME="E1:39_1_1_1"/>
( <font color="Maroon" title="c6">o</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c6">o</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E2:39_1_1_1"/>
( <font color="Maroon" title="c6">o</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c6">o</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:39_1_1"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c4">G</font>) <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">G</font>)
 <b>by </b><i><a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D21" target="_self" title="MONOID_0:def.21">MONOID_0:def 21</a><br/>

<b>hence </b><a NAME="E6:39_1_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">G</font>) is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">G</font>) is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">G</font>) is <a href="monoid_0.html#V8" title="MONOID_0:attr.8">cancelable</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">G</font>) is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">G</font>) is <a href="monoid_0.html#V5" title="MONOID_0:attr.5">invertible</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">G</font>) is <a href="monoid_0.html#V9" title="MONOID_0:attr.9">uniquely-decomposable</a> &amp;  ex <font color="Olive" title="b1">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font> st <font color="Olive" title="b1">e</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">G</font>) )
 <b>by </b><i><a class="ref" href="monoid_0.html#T3" target="_self" title="MONOID_0:th.3">Th3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a>,<a class="ref" href="monoid_0.html#D10" target="_self" title="MONOID_0:def.10">MONOID_0:def 10</a>,<a class="ref" href="monoid_0.html#D11" target="_self" title="MONOID_0:def.11">MONOID_0:def 11</a>,<a class="ref" href="monoid_0.html#D12" target="_self" title="MONOID_0:def.12">MONOID_0:def 12</a>,<a class="ref" href="monoid_0.html#D13" target="_self" title="MONOID_0:def.13">MONOID_0:def 13</a>,<a class="ref" href="monoid_0.html#D16" target="_self" title="MONOID_0:def.16">MONOID_0:def 16</a>,<a class="ref" href="monoid_0.html#D19" target="_self" title="MONOID_0:def.19">MONOID_0:def 19</a>,<a class="ref" href="monoid_0.html#D20" target="_self" title="MONOID_0:def.20">MONOID_0:def 20</a><br/>

<b>thus </b><a NAME="E7:39_1_1"/>
( ( for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font> holds  <font color="Olive" title="b1">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ) &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font> holds  <font color="Olive" title="b1">x</font> is   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ) )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D1" target="_self" title="MONOID_0:def.1">MONOID_0:def 1</a>,<a class="ref" href="monoid_0.html#D2" target="_self" title="MONOID_0:def.2">MONOID_0:def 2</a><br/>

<b>thus </b><a NAME="E8:39_1_1"/>
<font color="Maroon" title="c4">G</font> is <a href="algstr_0.html#V22" title="ALGSTR_0:attr.22">strict</a>
 ;<br/>


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