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<b>set </b><font color="Maroon" title="c2">a</font> = <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>);<br/>
<b>reconsider </b><font color="Maroon" title="c3">e</font> = <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>) as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>)</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>)</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c5">H</font> =  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">multLoopStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>)</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">e</font> #);<br/>
<a NAME="E3:65_1_1"/><i><font color="Green" title="E33">A1</font></i>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">multLoopStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>)</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">e</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">G</font>
 
;<br/>
<a NAME="E4:65_1_1"/>
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>)</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>) &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>) &amp; <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>) &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="monoid_0.html#T3" target="_self" title="MONOID_0:th.3">Th3</a>, <a class="ref" href="monoid_0.html#T18" target="_self" title="MONOID_0:th.18">Th18</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:65_1_1"/><b>then </b>
<font color="Maroon">H<sub>2</sub></font>( <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">multLoopStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>)</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">e</font> #)) <a href="funct_4.html#NR1" title="FUNCT_4:NR.1">&lt;=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>)
 
<b>by </b><i><a class="ref" href="funct_4.html#T8" title="FUNCT_4:th.8">FUNCT_4:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">H</font> =  <a href="vectsp_1.html#G1" title="VECTSP_1:aggr.1">multLoopStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>)</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c3">e</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="monoid_0.html#M3" title="MONOID_0:mode.3">MonoidalSubStr</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="monoid_0.html#E3:65_1_1"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="monoid_0.html#D25" target="_self" title="MONOID_0:def.25">Def25</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">H</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_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="c6">H</font>;<br/><a NAME="E1:65_1_1_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</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="E2:65_1_1_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">H</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c6">H</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">H<sub>2</sub></font>(<font color="Maroon" title="c6">H</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c6">H</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:65_1_1"/>
<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c6">H</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="c6">H</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>thus </b><a NAME="E9:65_1_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">H</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="c6">H</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="c6">H</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="c6">H</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="c6">H</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="c6">H</font>) is <a href="monoid_0.html#V9" title="MONOID_0:attr.9">uniquely-decomposable</a> )
 <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="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="E10:65_1_1"/>
<font color="Maroon" title="c6">H</font> is <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">strict</a>
 ;<br/>


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