<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">G</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ;<br/>

<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>
 ;<br/>

<b>thus </b><a NAME="E2:30"/>
( <font color="Maroon" title="c1">G</font> is <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a> implies ( <font color="Maroon" title="c1">G</font> is <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="finseqop.html#NR2" title="FINSEQOP:NR.2">has_an_inverseOp</a>  ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c2">e</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>
<b>consider </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</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="E4:30_1"/><i><font color="Green" title="E22">A3</font></i>: 
( <font color="Maroon" title="c2">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> &amp; <font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:30_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="monoid_0.html#T12" target="_self" title="MONOID_0:th.12">Th12</a></i>;<br/>
<div><i><font color="Green" title="E23">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/><b>consider </b><font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</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="E2:30_1_1"/><i><font color="Green" title="E24">A5</font></i>: 
( <font color="Maroon" title="c2">e</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c7">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:30_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="monoid_0.html#T12" target="_self" title="MONOID_0:th.12">Th12</a></i>;<br/><b>thus </b><a NAME="E3:30_1_1"/>
<font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:30"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="monoid_0.html#E4:30_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="monoid_0.html#E2:30_1_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="monoid_0.html#E28"><i><font color="Green" title="E14">Lm2</font></i></a></i>;<br/><b>thus </b><a NAME="E4:30_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:30"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="monoid_0.html#E4:30_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="monoid_0.html#E2:30_1_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="monoid_0.html#E28"><i><font color="Green" title="E14">Lm2</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:30_1"/><b>then </b><i><font color="Green" title="E24">A6</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )
 
;<br/>
<b>hence </b><a NAME="E7:30_1"/>
<font color="Maroon" title="c1">G</font> is <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>
 <b>by </b><i><a class="txt" href="monoid_0.html#E5:30_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="monoid_0.html#T6" target="_self" title="MONOID_0:th.6">Th6</a></i>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>] <b>means </b>( $1 <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; $2 <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> );<br/>
<a NAME="E8:30_1"/><i><font color="Green" title="E25">A7</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>consider </b><font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</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="E2:30_1_2"/><i><font color="Green" title="E26">A8</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c7">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E1:30_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="monoid_0.html#T12" target="_self" title="MONOID_0:th.12">Th12</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">b</font>
;<br/>

<a NAME="E3:30_1_2"/>
( <font color="Maroon" title="c7">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c7">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> )
 
<b>by </b><i><a class="txt" href="monoid_0.html#E1:30"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="monoid_0.html#E5:30_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="monoid_0.html#E6:30_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="monoid_0.html#E28"><i><font color="Green" title="E14">Lm2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:30_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>]
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:30_1_2"><i><font color="Green" title="E26">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:30_1"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> st <br/>for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="monoid_0.html#E8:30_1"><i><font color="Green" title="E25">A7</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c5">u</font> being   <a href="binop_1.html#NM1" title="BINOP_1:NM.1">UnOp</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>)<b> such that </b><br/><a NAME="E11:30_1"/><i><font color="Green" title="E26">A9</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>]
 ;<br/>
<b>take </b>
<font color="Maroon" title="c5">u</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="finseqop.html#D2" title="FINSEQOP:def.2">FINSEQOP:def 2</a><br/>

<b>let </b><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="finseqop.html#D1" title="FINSEQOP:def.1">FINSEQOP:def 1</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/><a NAME="E1:30_1_3"/>
( <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c7">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c3">x</font> )
 ;<br/><b>hence </b><a NAME="E2:30_1_3"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E5:30_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="monoid_0.html#E6:30_1"><i><font color="Green" title="E24">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:30_1"/><b>then </b>
<font color="Maroon" title="c3">x</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">G</font>)
 
<b>by </b><i><a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></i>;<br/>
<a NAME="E14:30_1"/><b>then </b>
( <font color="Maroon" title="c3">x</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>2</sub></font>(<font color="Maroon" title="c1">G</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span> &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font> )
 
<b>by </b><i><a class="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E15:30_1"/>
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <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>2</sub></font>(<font color="Maroon" title="c1">G</font>) &amp; <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c6">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>2</sub></font>(<font color="Maroon" title="c1">G</font>) )
 <b>by </b><i><a class="txt" href="monoid_0.html#E11:30_1"><i><font color="Green" title="E26">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:30"/><i><font color="Green" title="E21">A10</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D10" target="_self" title="MONOID_0:def.10">MONOID_0:def 10</a><br/>

<b>given </b><font color="Maroon" title="c2">u</font> being   <a href="binop_1.html#NM1" title="BINOP_1:NM.1">UnOp</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">G</font>)<b> such that </b><a NAME="E5:30"/><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c2">u</font> <a href="finseqop.html#R1" title="FINSEQOP:pred.1">is_an_inverseOp_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>)
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="finseqop.html#D2" title="FINSEQOP:def.2">FINSEQOP:def 2</a><br/>

<b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D5" target="_self" title="MONOID_0:def.5">MONOID_0:def 5</a>,<a class="ref" href="monoid_0.html#D16" target="_self" title="MONOID_0:def.16">MONOID_0:def 16</a><br/><b>let </b><font color="Maroon" title="c4">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>take </b><font color="Maroon" title="c5">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">c</font>;<br/>

<b>take </b><font color="Maroon" title="c6">r</font> = <font color="Maroon" title="c4">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font></span>)</span>;<br/>

<b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">l</font> = 
<font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">l</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E1:30"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="monoid_0.html#E28"><i><font color="Green" title="E14">Lm2</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>)</span>)</span>,<font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E5:30"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="finseqop.html#D1" title="FINSEQOP:def.1">FINSEQOP:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E3:30"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="setwiseo.html#T23" title="SETWISEO:th.23">SETWISEO:23</a></i>
;<br/>

<b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">r</font>,<font color="Maroon" title="c3">a</font> = 
<font color="Maroon" title="c6">r</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">a</font>

<br/>.= 
<font color="Maroon" title="c4">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">a</font></span>)</span>
<b>by </b><i><a class="txt" href="monoid_0.html#E1:30"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="monoid_0.html#E28"><i><font color="Green" title="E14">Lm2</font></i></a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>) <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">G</font>)</span>)</span>
<b>by </b><i><a class="txt" href="monoid_0.html#E5:30"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="finseqop.html#D1" title="FINSEQOP:def.1">FINSEQOP:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E3:30"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="setwiseo.html#T23" title="SETWISEO:th.23">SETWISEO:23</a></i>
;<br/>


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