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

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>,   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Olive">b<sub>2</sub></font> );<br/>
<a NAME="E1:136_1_1"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="partfun1.html#T120">PARTFUN1:120</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E3:136_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="binop_1.html#NM2">BinOp</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:136_1_1"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3">BINOP_1:sch 3</a>(<a class="txt" href="monoid_0.html#E1:136_1_1"><i><font color="Green">E62</font></i></a>);<br/></i>
<b>set </b><font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></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">c<sub>5</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></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">E64</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">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:136_1_1_3"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E3:136_1_1"><i><font color="Green">E63</font></i></a></i>;<br/><b>thus </b><a NAME="E3:136_1_1_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="monoid_0.html#E2:136_1_1_3"><i><font color="Green">E65</font></i></a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="ref" href="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<div><i><font color="Green">E65</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">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="partfun1.html#T120">PARTFUN1:120</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>7</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E6:136_1_1"><i><font color="Green">E64</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="partfun1.html#T37">PARTFUN1:37</a></i>
;<br/><b>thus </b><font color="Maroon">c<sub>8</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E6:136_1_1"><i><font color="Green">E64</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></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">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>11</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E6:136_1_1"><i><font color="Green">E64</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E6:136_1_1"><i><font color="Green">E64</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E6:136_1_1"><i><font color="Green">E64</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="monoid_0.html#E6:136_1_1"><i><font color="Green">E64</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></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><a class="txt" href="monoid_0.html#E9:136_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="monoid_0.html#E28"><i><font color="Green">Lemma14</font></i></a>, <a class="ref" href="monoid_0.html#T6" target="_self">Th6</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E12:136_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>8</sub></font>) <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>8</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E14:136_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>10</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<b>by </b><i><a class="txt" href="monoid_0.html#E3:136_1_1"><i><font color="Green">E63</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:136_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="monoid_0.html#K2" target="_self">[*]</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="monoid_0.html#NK2" target="_self">(*)</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>


</div>
