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<b>set </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>);<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) as    <a href="struct_0.html#M1">Element</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>)</span><a href="domain_1.html#K6">}</a></span> <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="binop_1.html#NM2">BinOp</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>)</span><a href="domain_1.html#K6">}</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="vectsp_1.html#G1">multLoopStr</a>(# <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>)</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> #);<br/>
<a NAME="E3:67_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>)</span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K6">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="group_1.html#K1">*</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="group_1.html#K1">*</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>)</span><a href="domain_1.html#K1">]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)</span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="monoid_0.html#T3" target="_self">Th3</a>, <a class="ref" href="monoid_0.html#T18" target="_self">Th18</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:67_1_1"/><b>then </b>
<font color="Maroon">H<sub>2</sub></font>( <a href="vectsp_1.html#G1">multLoopStr</a>(# <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>)</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> #)) <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 
<b>by </b><i><a class="ref" href="funct_4.html#T8">FUNCT_4:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="vectsp_1.html#G1">multLoopStr</a>(# <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>)</span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> #) as  non <a href="struct_0.html#V3">empty</a>  <a href="monoid_0.html#M3" target="_self">MonoidalSubStr</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="monoid_0.html#D25" target="_self">Def25</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E1:67_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E2:67_1_1_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>6</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>6</sub></font>),<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>6</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>6</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:67_1_1"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">c<sub>6</sub></font>) <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>6</sub></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>thus </b><a NAME="E8:67_1_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>6</sub></font>) is <a href="binop_1.html#V2">associative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>6</sub></font>) is <a href="binop_1.html#V1">commutative</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>6</sub></font>) is <a href="monoid_0.html#V8" target="_self">cancelable</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>6</sub></font>) is <a href="binop_1.html#V3">idempotent</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>6</sub></font>) is <a href="monoid_0.html#V5" target="_self">invertible</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>6</sub></font>) is <a href="monoid_0.html#V9" target="_self">uniquely-decomposable</a> )
 <b>by </b><i><a class="ref" href="monoid_0.html#T3" target="_self">Th3</a></i>; <i><font color="Red">:: according to </font></i><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="E9:67_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="vectsp_1.html#V1">strict</a>
 ;<br/>


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