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<b>consider </b><font color="Maroon">p</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">o</font> being   <a href="binop_1.html#NM2">BinOp</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">e</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<b>take </b><font color="Maroon">G</font> =  <a href="vectsp_1.html#G1">multLoopStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">o</font>,<font color="Maroon">e</font> #);<br/>

<b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> ;<br/>
<b>reconsider </b><font color="Maroon">o</font> = <font color="Maroon">o</font> as   <a href="monoid_0.html#NM1">BinOp</a> of <font color="Maroon">G</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><a NAME="E1:42_1_1_1"/>
( <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">e</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E2:42_1_1_1"/>
( <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">e</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">o</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:42_1_1"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">G</font>) <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>)
 <b>by </b><i><a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D21" target="_self">MONOID_0:def 21</a><br/>

<b>hence </b><a NAME="E6:42_1_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) is <a href="monoid_0.html#V8" target="_self">cancelable</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) is <a href="binop_1.html#V3">idempotent</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) is <a href="monoid_0.html#V5" target="_self">invertible</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) is <a href="monoid_0.html#V9" target="_self">uniquely-decomposable</a> &amp;  ex <font color="Olive">e</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <font color="Olive">e</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">G</font>) )
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a>,<a class="ref" href="monoid_0.html#D10" target="_self">MONOID_0:def 10</a>,<a class="ref" href="monoid_0.html#D11" target="_self">MONOID_0:def 11</a>,<a class="ref" href="monoid_0.html#D12" target="_self">MONOID_0:def 12</a>,<a class="ref" href="monoid_0.html#D13" target="_self">MONOID_0:def 13</a>,<a class="ref" href="monoid_0.html#D16" target="_self">MONOID_0:def 16</a>,<a class="ref" href="monoid_0.html#D19" target="_self">MONOID_0:def 19</a>,<a class="ref" href="monoid_0.html#D20" target="_self">MONOID_0:def 20</a><br/>

<b>thus </b><a NAME="E7:42_1_1"/>
( ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <font color="Olive">x</font> is   <a href="funct_1.html#NM1">Function</a> ) &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <font color="Olive">x</font> is   <a href="finseq_1.html#NM1">FinSequence</a> ) )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D1" target="_self">MONOID_0:def 1</a>,<a class="ref" href="monoid_0.html#D2" target="_self">MONOID_0:def 2</a><br/>

<b>thus </b><a NAME="E8:42_1_1"/>
<font color="Maroon">G</font> is <a href="vectsp_1.html#V1">strict</a>
 ;<br/>


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