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

<b>let </b><font color="Maroon" title="c1">Q</font> be   <a href="quantal1.html#NM2" title="QUANTAL1:NM.2">Quantale</a>;<br/><b>let </b><font color="Maroon" title="c2">D</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>;<br/>



<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E22">A1</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">Q</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D17" target="_self" title="QUANTAL1:def.17">QUANTAL1:def 17</a><br/>

<b>set </b><font color="Maroon" title="c3">I</font> = <font color="Maroon" title="c2">D</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font>;<br/>
<b>set </b><font color="Maroon" title="c4">J</font> = <font color="Maroon" title="c2">D</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font>;<br/>
<div><i><font color="Green" title="E23">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_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">Q</font>;<br/><b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $1;<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">Q</font>] <b>means </b>$1 <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">D</font>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>] <b>means </b>$1 <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">D</font>;<br/><a NAME="E1:48_1"/><i><font color="Green" title="E24">A3</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">b</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>;<br/>

<a NAME="E1:48_1_1"/>
( <font color="Maroon" title="c6">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="txt" href="quantal1.html#E1:48"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E2:48_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">b</font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">b</font>] )
 <b>by </b><i><a class="ref" href="quantal1.html#T12" target="_self" title="QUANTAL1:th.12">Th12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E2:48_1"/><i><font color="Green" title="E25">A4</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">c</font>) where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">c</font>] </span>}</span> 
 <b>from </b><i><a class="ref" href="fraenkel.html#S3" title="FRAENKEL:sch.3">FRAENKEL:sch 3</a>(<a class="txt" href="quantal1.html#E1:48_1"><i><font color="Green" title="E24">A3</font></i></a>);<br/></i><b>thus </b><font color="Maroon" title="c5">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font>
<b>by </b><i><a class="txt" href="quantal1.html#E1:48"><i><font color="Green" title="E22">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font>
<b>by </b><i><a class="txt" href="quantal1.html#E2:48_1"><i><font color="Green" title="E25">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="txt" href="quantal1.html#E1:48"><i><font color="Green" title="E22">A1</font></i></a></i>
;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>] <b>means </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> $1 <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">D</font>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>] <b>means </b><font color="Maroon" title="c5">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> $1 <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">D</font>;<br/><a NAME="E4:48_1"/><i><font color="Green" title="E26">A5</font></i>: 
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> holds <br/> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">b</font>] iff <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">b</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>;<br/>

<a NAME="E1:48_1_3"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c6">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="txt" href="quantal1.html#E1:48"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E2:48_1_3"/>
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c6">b</font>] iff <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c6">b</font>] )
 <b>by </b><i><a class="ref" href="quantal1.html#T11" target="_self" title="QUANTAL1:th.11">Th11</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:48_1"/><i><font color="Green" title="E27">A6</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">b</font>) where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">b</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b1">c</font>) where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">c</font>] </span>}</span> 
 <b>from </b><i><a class="ref" href="fraenkel.html#S3" title="FRAENKEL:sch.3">FRAENKEL:sch 3</a>(<a class="txt" href="quantal1.html#E4:48_1"><i><font color="Green" title="E26">A5</font></i></a>);<br/></i><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font>
<b>by </b><i><a class="txt" href="quantal1.html#E1:48"><i><font color="Green" title="E22">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font>
<b>by </b><i><a class="txt" href="quantal1.html#E5:48_1"><i><font color="Green" title="E27">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="txt" href="quantal1.html#E1:48"><i><font color="Green" title="E22">A1</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E3:48"/><b>then </b><i><font color="Green" title="E24">A7</font></i>: 
( <font color="Maroon" title="c2">D</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> &amp; <font color="Maroon" title="c2">D</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> )
 
;<br/>
<a NAME="E4:48"/><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c2">D</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c1">Q</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:48_2"/>
<font color="Maroon" title="c2">D</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c1">Q</font>)
  <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D7" title="BINOP_1:def.7">BINOP_1:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_1"><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">Q</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D16" title="BINOP_1:def.16">BINOP_1:def 16</a><br/>

<b>thus </b><font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c1">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span>,<font color="Maroon" title="c5">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="txt" href="quantal1.html#E3:48"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="group_1.html#D1" title="GROUP_1:def.1">GROUP_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="txt" href="quantal1.html#E2:48"><i><font color="Green" title="E23">A2</font></i></a></i>
;<br/>


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

<b>thus </b><a NAME="E2:48_2"/>
<font color="Maroon" title="c2">D</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font> <a href="binop_1.html#R2" title="BINOP_1:pred.2">is_a_right_unity_wrt</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c1">Q</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_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">Q</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D17" title="BINOP_1:def.17">BINOP_1:def 17</a><br/>

<b>thus </b><font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c1">Q</font>) <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span> = 
<font color="Maroon" title="c5">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D1" title="GROUP_1:def.1">GROUP_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="txt" href="quantal1.html#E2:48"><i><font color="Green" title="E23">A2</font></i></a></i>
;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:48"/>
<font color="Maroon">H<sub>4</sub></font>(<font color="Maroon" title="c1">Q</font>) <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a> 
 <b>by </b><i><a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D10" title="MONOID_0:def.10">MONOID_0:def 10</a><br/>

<b>thus </b><a NAME="E6:48"/>
(  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> <font color="Maroon" title="c2">D</font> &amp;  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> <font color="Maroon" title="c2">D</font> )
 <b>by </b><i><a class="txt" href="quantal1.html#E3:48"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="quantal1.html#E4:48"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="binop_1.html#D8" title="BINOP_1:def.8">BINOP_1:def 8</a></i>;<br/>


</div>
