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

<b>set </b><font color="Maroon">D</font> =  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>,   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>,   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>] <b>means </b> ex <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Olive">g</font> );<br/>
<a NAME="E1:136_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>  ex <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>, <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/>

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">a</font>, <font color="Maroon">g</font> = <font color="Maroon">b</font> as   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> <b>by </b><i><a class="ref" href="partfun1.html#T120">PARTFUN1:120</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<b>take </b>
<font color="Maroon">c</font>
;<br/>

<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>thus </b><a NAME="E3:136_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">g</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="binop_1.html#NM2">BinOp</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:136_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">a</font>,<font color="Olive">b</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>(<a class="ref" href="monoid_0.html#T16" target="_self">Th16</a>);<br/></i>
<b>set </b><font color="Maroon">G</font> =  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<font color="Maroon">f</font> #);<br/>
<a NAME="E4:136_1_1"/>
 <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<font color="Maroon">f</font> #) is <a href="monoid_0.html#V1" target="_self">constituted-Functions</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<font color="Maroon">f</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D1" target="_self">MONOID_0:def 1</a><br/>

<b>thus </b><a NAME="E1:136_1_1_2"/>
<font color="Maroon">x</font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i><a class="ref" href="partfun1.html#T120">PARTFUN1:120</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G</font> =  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font></span>)</span>,<font color="Maroon">f</font> #) as  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V1">strict</a> <a href="monoid_0.html#V1" target="_self">constituted-Functions</a>  <a href="group_1.html#L1">HGrStr</a>  ;<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font>, <font color="Maroon">h</font> being   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:136_1_1_3"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">h</font> )
 <b>by </b><i><a class="ref" href="monoid_0.html#T17" target="_self">Th17</a></i>;<br/><b>thus </b><a NAME="E3:136_1_1_3"/>
<font color="Maroon">a</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">g</font> =  <a href="partfun1.html#K6">id</a> <font color="Maroon">X</font> as   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">h</font> as   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> <b>by </b><i><a class="ref" href="partfun1.html#T120">PARTFUN1:120</a></i>;<br/><b>thus </b><font color="Maroon">f1</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">h</font> = 
<font color="Maroon">g</font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">j</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="partfun1.html#T37">PARTFUN1:37</a></i>
;<br/><b>thus </b><font color="Maroon">h</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">f1</font> = 
<font color="Maroon">j</font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">g</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="partfun1.html#T36">PARTFUN1:36</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>reconsider </b><font color="Maroon">fg</font> = <font color="Maroon">f'</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">g</font>, <font color="Maroon">gh</font> = <font color="Maroon">g</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">h</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">g</font></span>)</span> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">h</font> = 
<font color="Maroon">fg</font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">h</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">g</font></span>)</span> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">h</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f'</font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">f'</font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">gh</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f'</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V1">strict</a> <a href="group_1.html#V2">unital</a> <a href="group_1.html#V4">associative</a> <a href="monoid_0.html#V1" target="_self">constituted-Functions</a>  <a href="group_1.html#L1">HGrStr</a>  <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>thus </b><a NAME="E12:136_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G</font>) <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font>
 ;<br/>

<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>

<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font>, <font color="Maroon">h'</font> = <font color="Maroon">h</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">X</font>,<font color="Maroon">X</font> ;<br/>
<a NAME="E14:136_1_1"/>
( <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">g</font>,<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">h</font> &amp;  ex <font color="Olive">g</font>, <font color="Olive">h</font> being  <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font> st <br/>( <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp; <font color="Maroon">h'</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">g'</font>,<font color="Maroon">h'</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Olive">h</font> ) )
 
<b>by </b><i><a class="ref" href="monoid_0.html#T17" target="_self">Th17</a></i>;<br/>
<b>hence </b><a NAME="E15:136_1_1"/>
<font color="Maroon">g</font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">h</font>
 ;<br/>


</div>
