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<a NAME="E1:39_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a> 
 
<b>by </b><i><a class="ref" href="monoid_0.html#D10" title="MONOID_0:def.10">MONOID_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:39_1_1"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c3">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>)
 <b>by </b><i><a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">M</font> =  <a href="algstr_0.html#G4" title="ALGSTR_0:aggr.4">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">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="E4:39_1_1"/><i><font color="Green" title="E22">A2</font></i>: 
 <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="fraenkel.html#K1" title="FRAENKEL:func.1">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>
 
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<a NAME="E5:39_1_1"/><i><font color="Green" title="E23">A3</font></i>: 
( <font color="Maroon" title="c3">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;  <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="fraenkel.html#K1" title="FRAENKEL:func.1">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> #) )
 
<b>by </b><i><a class="txt" href="monoid_1.html#E3:39_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="monoid_1.html#D3" target="_self" title="MONOID_1:def.3">Def3</a>, <a class="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</a></i>;<br/>
<a NAME="E6:39_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="monoid_1.html#K5" title="MONOID_1:func.5">--&gt;</a> <font color="Maroon" title="c3">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>
 
<b>by </b><i><a class="txt" href="monoid_1.html#E3:39_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="monoid_1.html#T10" target="_self" title="MONOID_1:th.10">Th10</a></i>;<br/>
<b>hence </b><a NAME="E7:39_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> 
 <b>by </b><i><a class="txt" href="monoid_1.html#E4:39_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="monoid_1.html#E5:39_1_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="monoid_0.html#D21" title="MONOID_0:def.21">MONOID_0:def 21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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