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

<b>set </b><font color="Maroon" title="c2">D</font> =  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>] <b>means </b> ex <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Olive" title="b2">g</font> );<br/>
<a NAME="E1:132_1_1"/><i><font color="Green" title="E58">A1</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Olive" title="b1">c</font>]</span><br/>

<b>reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c4">b</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="partfun1.html#T120" title="PARTFUN1:th.120">PARTFUN1:120</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">c</font> = <font color="Maroon" title="c5">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">g</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">c</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">c</font>]</span><br/>

<b>take </b>
<font color="Maroon" title="c5">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Olive" title="b1">g</font> )</span><br/>

<b>take </b>
<font color="Maroon" title="c6">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c6">g</font> )</span><br/>

<b>thus </b><a NAME="E3:132_1_1_1"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c6">g</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">f</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:132_1_1"/><i><font color="Green" title="E59">A2</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="monoid_0.html#E1:132_1_1"><i><font color="Green" title="E58">A1</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c4">G</font> =  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c3">f</font> #);<br/>
<a NAME="E4:132_1_1"/>
 <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c3">f</font> #) is <a href="monoid_0.html#V1" title="MONOID_0:attr.1">constituted-Functions</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c3">f</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D1" target="_self" title="MONOID_0:def.1">MONOID_0:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/>

<b>thus </b><a NAME="E1:132_1_1_2"/>
<font color="Maroon" title="c5">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 <b>by </b><i><a class="ref" href="partfun1.html#T120" title="PARTFUN1:th.120">PARTFUN1:120</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">G</font> =  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c3">f</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a> <a href="monoid_0.html#V1" title="MONOID_0:attr.1">constituted-Functions</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  ;<br/>
<div><i><font color="Green" title="E60">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">a</font>, <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="c5">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c7">b</font></span><br/><b>consider </b><font color="Maroon" title="c8">g</font>, <font color="Maroon" title="c9">h</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E2:132_1_1_3"/><i><font color="Green" title="E61">A4</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h</font> &amp; <font color="Maroon" title="c3">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c9">h</font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E3:132_1_1"><i><font color="Green" title="E59">A2</font></i></a></i>;<br/><b>thus </b><a NAME="E3:132_1_1_3"/>
<font color="Maroon" title="c6">a</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:132_1_1_3"><i><font color="Green" title="E61">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c6">g</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">X</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">f1</font> = <font color="Maroon" title="c6">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">G</font> <b>by </b><i><a class="ref" href="partfun1.html#T119" title="PARTFUN1:th.119">PARTFUN1:119</a></i>;<br/>
<div><i><font color="Green" title="E61">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">f1</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> &amp; <font color="Maroon" title="c8">h</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c9">j</font> = <font color="Maroon" title="c8">h</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="partfun1.html#T120" title="PARTFUN1:th.120">PARTFUN1:120</a></i>;<br/><b>thus </b><font color="Maroon" title="c7">f1</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c8">h</font> = 
<font color="Maroon" title="c6">g</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c9">j</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E6:132_1_1"><i><font color="Green" title="E60">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">h</font>
<b>by </b><i><a class="ref" href="partfun1.html#T37" title="PARTFUN1:th.37">PARTFUN1:37</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">h</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font></span><br/><b>thus </b><font color="Maroon" title="c8">h</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c7">f1</font> = 
<font color="Maroon" title="c9">j</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c6">g</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E6:132_1_1"><i><font color="Green" title="E60">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">h</font>
<b>by </b><i><a class="ref" href="partfun1.html#T36" title="PARTFUN1:th.36">PARTFUN1:36</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">f'</font>, <font color="Maroon" title="c9">g</font>, <font color="Maroon" title="c10">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f'</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f'</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c10">h</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c11">fg</font> = <font color="Maroon" title="c8">f'</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c9">g</font>, <font color="Maroon" title="c12">gh</font> = <font color="Maroon" title="c9">g</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c10">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">G</font> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c8">f'</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c10">h</font> = 
<font color="Maroon" title="c11">fg</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c10">h</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E6:132_1_1"><i><font color="Green" title="E60">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f'</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c10">h</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E6:132_1_1"><i><font color="Green" title="E60">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">f'</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c10">h</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon" title="c8">f'</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c12">gh</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E6:132_1_1"><i><font color="Green" title="E60">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">f'</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c10">h</font></span>)</span>
<b>by </b><i><a class="txt" href="monoid_0.html#E6:132_1_1"><i><font color="Green" title="E60">A3</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">G</font> = <font color="Maroon" title="c5">G</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="monoid_0.html#V1" title="MONOID_0:attr.1">constituted-Functions</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  <b>by </b><i><a class="txt" href="monoid_0.html#E9:132_1_1"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="monoid_0.html#E28"><i><font color="Green" title="E14">Lm2</font></i></a>, <a class="ref" href="monoid_0.html#T6" target="_self" title="MONOID_0:th.6">Th6</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">G</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">G</font> holds  <font color="Olive" title="b1">f</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Olive" title="b2">g</font> ) )</span><br/>

<b>thus </b><a NAME="E12:132_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">G</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">G</font> holds  <font color="Olive" title="b1">f</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Olive" title="b2">g</font></span><br/>

<b>let </b><font color="Maroon" title="c9">g</font>, <font color="Maroon" title="c10">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">g</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c10">h</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">g'</font> = <font color="Maroon" title="c9">g</font>, <font color="Maroon" title="c12">h'</font> = <font color="Maroon" title="c10">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E14:132_1_1"/>
( <font color="Maroon" title="c3">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c10">h</font> &amp;  ex <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c11">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c12">h'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> &amp; <font color="Maroon" title="c3">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c11">g'</font>,<font color="Maroon" title="c12">h'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Olive" title="b2">h</font> ) )
 
<b>by </b><i><a class="txt" href="monoid_0.html#E3:132_1_1"><i><font color="Green" title="E59">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:132_1_1"/>
<font color="Maroon" title="c9">g</font> <a href="monoid_0.html#NK3" title="MONOID_0:NK.3">[*]</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="monoid_0.html#NK2" title="MONOID_0:NK.2">(*)</a> <font color="Maroon" title="c10">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
