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<span class="kw">set </span><font color="Maroon" title="c3">M</font> =  <a href="algstr_0.html#G4" title="ALGSTR_0:aggr.4">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>)</span>)</span></span>)</span> #);<br/>
<a NAME="E1:40_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>) is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D10" title="MONOID_0:def.10">MONOID_0:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E3:40_1_1"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a></span>;<br/>
<a NAME="E4:40_1_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
(  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="algstr_0.html#G4" title="ALGSTR_0:aggr.4">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>)</span>)</span></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">G</font></span>)</span> &amp;  <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G4" title="ALGSTR_0:aggr.4">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <span class="p2">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>)</span>)</span></span>)</span> #) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_1.html#D3" target="_self" title="MONOID_1:def.3">Def3</a></span>;<br/>
<a NAME="E5:40_1_1"/>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>) &amp; <font color="Maroon" title="c2">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <font color="Maroon" title="c4">a</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="monoid_1.html#K8" title="MONOID_1:func.8">.:</a> <font color="Maroon" title="c2">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_1"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="monoid_1.html#T10" target="_self" title="MONOID_1:th.10">Th10</a>, <a class="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:40_1_1"/>
 <a href="monoid_1.html#K9" title="MONOID_1:func.9">.:</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">A</font> is  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V22" title="ALGSTR_0:attr.22">strict</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="monoid_0.html#V1" title="MONOID_0:attr.1">constituted-Functions</a>   <a href="algstr_0.html#L4" title="ALGSTR_0:struct.4">multLoopStr</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="monoid_0.html#D21" title="MONOID_0:def.21">MONOID_0:def 21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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