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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="monoid_0.html#K10" target="_self">*+^</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="monoid_0.html#K10" target="_self">*+^</a> );<br/>
<a NAME="E1:128"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="monoid_0.html#K10" target="_self">*+^</a> ) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="monoid_0.html#K10" target="_self">*+^</a> </span>)</span> holds <font color="Olive">b<sub>1</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K7">^</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<b>by </b><i><a class="ref" href="monoid_0.html#D34" target="_self">Def34</a></i>;<br/>
<a NAME="E2:128"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="monoid_0.html#K10" target="_self">*+^</a> </span>)</span> <b>by </b><i><a class="txt" href="monoid_0.html#E1:128"><i><font color="Green">E61</font></i></a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="monoid_0.html#K10" target="_self">*+^</a> </span>)</span>;<br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="monoid_0.html#K10" target="_self">*+^</a> ) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>5</sub></font>

<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="monoid_0.html#D34" target="_self">Def34</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
;<br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="monoid_0.html#K10" target="_self">*+^</a> ) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="monoid_0.html#D34" target="_self">Def34</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E5:128"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="monoid_0.html#K10" target="_self">*+^</a> )
 
<b>by </b><i><a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/>
<b>hence </b><a NAME="E6:128"/>
 <a href="binop_1.html#K3">the_unity_wrt</a> the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="monoid_0.html#K10" target="_self">*+^</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/>


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