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

<b>set </b><font color="Maroon">Ob3</font> = <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K3">:]</a></span>;<br/>
<b>set </b><font color="Maroon">G</font> =  <a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>consider </b><font color="Maroon">o</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:24_1_1"/><i><font color="Green">E16</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K3">:]</a></span>
 <b>and </b><br/><a NAME="E3:24_1_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K3">:]</a></span> holds <br/><font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">e</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">o</font> = <font color="Maroon">o</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K3">:]</a></span> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:24_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">o</font>
 ;<br/><a NAME="E2:24_1_1_1"/><b>then </b>
<font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:24_1_1_1"/>
<font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">o</font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K3">:]</a></span> <b>by </b><i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:24_1_1_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K3">:]</a></span>
 ;<br/><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><b>consider </b><font color="Maroon">i</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">j</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">k</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:24_1_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="domain_1.html#T15">DOMAIN_1:15</a></i>;<br/><b>reconsider </b><font color="Maroon">e'</font> = <font color="Maroon">e</font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K3">:]</a></span> <b>by </b><i/>;<br/><i><font color="Green">E28</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a>  = 
<font color="Maroon">e'</font> <a href="mcart_1.html#K5">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T50">MCART_1:50</a></i>
<br/>.= 
<font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T47">MCART_1:47</a></i>
;<br/><i><font color="Green">E29</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a>  = 
<font color="Maroon">e'</font> <a href="mcart_1.html#K6">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T50">MCART_1:50</a></i>
<br/>.= 
<font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T47">MCART_1:47</a></i>
;<br/><i><font color="Green">E30</font></i>: <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  = 
<font color="Maroon">e'</font> <a href="mcart_1.html#K7">`3</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T50">MCART_1:50</a></i>
<br/>.= 
<font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#T47">MCART_1:47</a></i>
;<br/><a NAME="E9:24_1_1_2"/><i><font color="Green">E31</font></i>: 
( <span class="p1">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">j</font>,<font color="Maroon">k</font> )
 <b>by </b><i/>;<br/><a NAME="E10:24_1_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , , , , </i>;<br/><i><font color="Green">E33</font></i>: <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> = 
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="altcat_1.html#D5">ALTCAT_1:def 5</a></i>
<br/>.= 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">i</font>,<font color="Maroon">k</font>
<b>by </b><i/>
;<br/><i><font color="Green">E35</font></i>: <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> = 
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span></span><a href="altcat_1.html#K3">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
<b>by </b><i><a class="txt" href="altcat_2.html#E2:24_1_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="altcat_1.html#D6">ALTCAT_1:def 6</a></i>
;<br/><div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_1_2_6"/>
<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:24_1_1_2_6"/><b>then </b>
(  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="ref" href="cat_1.html#T52">CAT_1:52</a></i>;<br/><b>hence </b><a NAME="E3:24_1_1_2_6"/>
<span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/></div><b>end;</b></div><a NAME="E14:24_1_1_2"/>
<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>
 <b>by </b><i/>;<br/><a NAME="E15:24_1_1_2"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="altcat_2.html#E2:24_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="altcat_2.html#E3:24_1_1"><i><font color="Green">E17</font></i></a>, , <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E16:24_1_1_2"/>
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="relat_1.html#K9">.:</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">i</font>,<font color="Maroon">k</font>
 <b>by </b><i/>;<br/><a NAME="E17:24_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="altcat_2.html#E2:24_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="altcat_2.html#E3:24_1_1"><i><font color="Green">E17</font></i></a>, , <a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>;<br/><b>hence </b><a NAME="E18:24_1_1_2"/>
<font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="altcat_1.html#K3">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span></span><a href="altcat_1.html#K3">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="altcat_2.html#E1:24_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="altcat_2.html#E4:24_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">o</font> as   <a href="altcat_1.html#NM3">BinComp</a> of <span class="p1">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>take </b>
<font color="Maroon">o</font>
;<br/>

<b>let </b><font color="Maroon">i</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">j</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>, <font color="Maroon">k</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/>

<a NAME="E9:24_1_1"/><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="domain_1.html#T15">DOMAIN_1:15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e</font> = <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K3">:]</a></span> ;<br/>
<i><font color="Green">E39</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a>  = 
<font color="Maroon">e</font> <a href="mcart_1.html#K5">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T50">MCART_1:50</a></i>
<br/>.= 
<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T47">MCART_1:47</a></i>
;<br/>
<i><font color="Green">E40</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a>  = 
<font color="Maroon">e</font> <a href="mcart_1.html#K6">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T50">MCART_1:50</a></i>
<br/>.= 
<font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T47">MCART_1:47</a></i>
;<br/>
<i><font color="Green">E41</font></i>: <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  = 
<font color="Maroon">e</font> <a href="mcart_1.html#K7">`3</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T50">MCART_1:50</a></i>
<br/>.= 
<font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T47">MCART_1:47</a></i>
;<br/>
<b>thus </b><font color="Maroon">o</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font> = 
<font color="Maroon">o</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span><a href="domain_1.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" target="_self">the_hom_sets_of</a> <font color="Maroon">C</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, , , , </i>
;<br/>


</div>
