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<b>set </b><font color="Maroon" title="c2">M</font> =  <a href="algstr_0.html#G4" title="ALGSTR_0:aggr.4">multLoopStr</a>(# <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <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>) <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><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><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> #);<br/>
<a NAME="E1:86_1_1"/><i><font color="Green" title="E51">A1</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="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <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>) <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><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><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> #) <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">(<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><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
;<br/>
<a NAME="E2:86_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/>
<a NAME="E3:86_1_1"/><b>then </b>
(  <a href="monoid_1.html#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</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="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <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>) <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a> </span>)</span>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><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><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> #) &amp; <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><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><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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>) <a href="monoid_1.html#K21" title="MONOID_1:func.21">.:^2</a>  )
 
<b>by </b><i><a class="ref" href="monoid_1.html#D9" target="_self" title="MONOID_1:def.9">Def9</a>, <a class="ref" href="monoid_1.html#T54" target="_self" title="MONOID_1:th.54">Th54</a></i>;<br/>
<b>hence </b><a NAME="E4:86_1_1"/>
 <a href="monoid_1.html#K22" title="MONOID_1:func.22">bool</a> <font color="Maroon" title="c1">G</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="algstr_0.html#L4" title="ALGSTR_0:struct.4">multLoopStr</a> 
 <b>by </b><i><a class="txt" href="monoid_1.html#E1:86_1_1"><i><font color="Green" title="E51">A1</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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