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<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>set </b><font color="Maroon" title="c2">N</font> = <font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K9" title="MONOID_0:func.9">*+^</a> ;<br/>
<b>set </b><font color="Maroon" title="c3">f</font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K9" title="MONOID_0:func.9">*+^</a> );<br/>
<a NAME="E1:124"/><i><font color="Green" title="E57">A1</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K9" title="MONOID_0:func.9">*+^</a> ) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; ( for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K9" title="MONOID_0:func.9">*+^</a> </span>)</span> holds  <font color="Olive" title="b1">p</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b2">q</font> ) )
 
<b>by </b><i><a class="ref" href="monoid_0.html#D34" target="_self" title="MONOID_0:def.34">Def34</a></i>;<br/>
<a NAME="E2:124"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">D</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">a</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K9" title="MONOID_0:func.9">*+^</a> </span>)</span> <b>by </b><i><a class="txt" href="monoid_0.html#E1:124"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="124_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K9" title="MONOID_0:func.9">*+^</a> </span>)</span>;<br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K9" title="MONOID_0:func.9">*+^</a> ) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> = 
<font color="Maroon" title="c4">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">b</font>

<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="ref" href="monoid_0.html#D34" target="_self" title="MONOID_0:def.34">Def34</a></i>
<br/>.= 
<font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>
;<br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K9" title="MONOID_0:func.9">*+^</a> ) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> = 
<font color="Maroon" title="c5">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c4">a</font>

<br/>.= 
<font color="Maroon" title="c5">b</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="monoid_0.html#D34" target="_self" title="MONOID_0:def.34">Def34</a></i>
<br/>.= 
<font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E5:124"/><b>then </b>
<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>2</sub></font>(<font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K9" title="MONOID_0:func.9">*+^</a> )
 
<b>by </b><i><a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></i>;<br/>
<b>hence </b><a NAME="E6:124"/>
 <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 <span class="p1">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="monoid_0.html#K9" title="MONOID_0:func.9">*+^</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</a></i>;<br/>


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