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

<b>set </b><font color="Maroon" title="c2">Ob3</font> = <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c3">G</font> =  <a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</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>  = the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>,<span class="p3">(<span class="default">$1 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<b>consider </b><font color="Maroon" title="c4">o</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:24_1_1"/><i><font color="Green" title="E9">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 <b>and </b><br/><a NAME="E3:24_1_1"/><i><font color="Green" title="E10">A2</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> holds <br/><font color="Maroon" title="c4">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">e</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c5">o</font> = <font color="Maroon" title="c4">o</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> <b>by </b><i><a class="txt" href="altcat_2.html#E2:24_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">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" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:24_1_1_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">o</font>
 ;<br/><a NAME="E2:24_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">e</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="altcat_2.html#E2:24_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="altcat_2.html#E3:24_1_1"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:24_1_1_1"/>
<font color="Maroon" title="c5">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 ;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">o</font> = <font color="Maroon" title="c5">o</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> <b>by </b><i><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">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" title="c7">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:24_1_1_2"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c6">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><b>consider </b><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font>, <font color="Maroon" title="c11">k</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E4:24_1_1_2"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="altcat_2.html#E1:24_1_1_2"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="domain_1.html#T15" title="DOMAIN_1:th.15">DOMAIN_1:15</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">e'</font> = <font color="Maroon" title="c7">e</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> <b>by </b><i><a class="txt" href="altcat_2.html#E1:24_1_1_2"><i><font color="Green" title="E11">A3</font></i></a></i>;<br/><i><font color="Green" title="E13">A5</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  = 
<font color="Maroon" title="c12">e'</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:24_1_1_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="mcart_1.html#T50" title="MCART_1:th.50">MCART_1:50</a></i>
<br/>.= 
<font color="Maroon" title="c9">i</font>
<b>by </b><i><a class="txt" href="altcat_2.html#E4:24_1_1_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="mcart_1.html#T47" title="MCART_1:th.47">MCART_1:47</a></i>
;<br/><i><font color="Green" title="E14">A6</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  = 
<font color="Maroon" title="c12">e'</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:24_1_1_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="mcart_1.html#T50" title="MCART_1:th.50">MCART_1:50</a></i>
<br/>.= 
<font color="Maroon" title="c10">j</font>
<b>by </b><i><a class="txt" href="altcat_2.html#E4:24_1_1_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="mcart_1.html#T47" title="MCART_1:th.47">MCART_1:47</a></i>
;<br/><i><font color="Green" title="E15">A7</font></i>: <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  = 
<font color="Maroon" title="c12">e'</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
<b>by </b><i><a class="txt" href="altcat_2.html#E4:24_1_1_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="mcart_1.html#T50" title="MCART_1:th.50">MCART_1:50</a></i>
<br/>.= 
<font color="Maroon" title="c11">k</font>
<b>by </b><i><a class="txt" href="altcat_2.html#E4:24_1_1_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="mcart_1.html#T47" title="MCART_1:th.47">MCART_1:47</a></i>
;<br/><a NAME="E9:24_1_1_2"/><i><font color="Green" title="E16">A8</font></i>: 
( <span class="p1">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> &amp; <span class="p1">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font> )
 <b>by </b><i><a class="ref" href="altcat_2.html#D3" target="_self" title="ALTCAT_2:def.3">Def3</a></i>;<br/><a NAME="E10:24_1_1_2"/><i><font color="Green" title="E17">A9</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="altcat_2.html#E3:24_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="altcat_2.html#E4:24_1_1_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="altcat_2.html#E6:24_1_1_2"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="altcat_2.html#E7:24_1_1_2"><i><font color="Green" title="E14">A6</font></i></a>, <a class="txt" href="altcat_2.html#E8:24_1_1_2"><i><font color="Green" title="E15">A7</font></i></a></i>;<br/><i><font color="Green" title="E18">A10</font></i>: <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> = 
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font>
<b>by </b><i><a class="txt" href="altcat_2.html#E4:24_1_1_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">k</font>
<b>by </b><i><a class="ref" href="altcat_1.html#D5" title="ALTCAT_1:def.5">ALTCAT_1:def 5</a></i>
<br/>.= 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">k</font>
<b>by </b><i><a class="ref" href="altcat_2.html#D3" target="_self" title="ALTCAT_2:def.3">Def3</a></i>
;<br/><i><font color="Green" title="E19">A11</font></i>: <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> = 
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font>
<b>by </b><i><a class="txt" href="altcat_2.html#E4:24_1_1_2"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
<b>by </b><i><a class="txt" href="altcat_2.html#E9:24_1_1_2"><i><font color="Green" title="E16">A8</font></i></a>, <a class="ref" href="altcat_1.html#D6" title="ALTCAT_1:def.6">ALTCAT_1:def 6</a></i>
;<br/><div><i><font color="Green" title="E20">A12</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" title="ALTCAT_1:func.2">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E2:24_1_1_2_6"/><b>then </b>
(  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="altcat_2.html#E11:24_1_1_2"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="cat_1.html#T52" title="CAT_1:th.52">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" title="ALTCAT_1:func.3">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="altcat_2.html#E12:24_1_1_2"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">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" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="altcat_2.html#T11" target="_self" title="ALTCAT_2:th.11">Th11</a></i>;<br/><a NAME="E15:24_1_1_2"/><b>then </b><i><font color="Green" title="E21">A13</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font>
 <b>by </b><i><a class="txt" href="altcat_2.html#E9:24_1_1_2"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="altcat_2.html#E10:24_1_1_2"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="altcat_2.html#E12:24_1_1_2"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E16:24_1_1_2"/>
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c10">j</font>,<font color="Maroon" title="c11">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">k</font>
 <b>by </b><i><a class="ref" href="altcat_2.html#T12" target="_self" title="ALTCAT_2:th.12">Th12</a></i>;<br/><a NAME="E17:24_1_1_2"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font>
 <b>by </b><i><a class="txt" href="altcat_2.html#E9:24_1_1_2"><i><font color="Green" title="E16">A8</font></i></a>, <a class="txt" href="altcat_2.html#E10:24_1_1_2"><i><font color="Green" title="E17">A9</font></i></a>, <a class="txt" href="altcat_2.html#E11:24_1_1_2"><i><font color="Green" title="E18">A10</font></i></a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>;<br/><b>hence </b><a NAME="E18:24_1_1_2"/>
<font color="Maroon" title="c6">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font>,<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font>
 <b>by </b><i><a class="txt" href="altcat_2.html#E13:24_1_1_2"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="altcat_2.html#E15:24_1_1_2"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">o</font> = <font color="Maroon" title="c6">o</font> as   <a href="altcat_1.html#NM3" title="ALTCAT_1:NM.3">BinComp</a> of <span class="p1">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <b>by </b><i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">o</font>
;<br/>

<b>let </b><font color="Maroon" title="c8">i</font>, <font color="Maroon" title="c9">j</font>, <font color="Maroon" title="c10">k</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c11">e</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font>,the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> ;<br/>
<i><font color="Green" title="E11">A14</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  = 
<font color="Maroon" title="c11">e</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T50" title="MCART_1:th.50">MCART_1:50</a></i>
<br/>.= 
<font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T47" title="MCART_1:th.47">MCART_1:47</a></i>
;<br/>
<i><font color="Green" title="E12">A15</font></i>: <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  = 
<font color="Maroon" title="c11">e</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T50" title="MCART_1:th.50">MCART_1:50</a></i>
<br/>.= 
<font color="Maroon" title="c9">j</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T47" title="MCART_1:th.47">MCART_1:47</a></i>
;<br/>
<i><font color="Green" title="E13">A16</font></i>: <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  = 
<font color="Maroon" title="c11">e</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T50" title="MCART_1:th.50">MCART_1:50</a></i>
<br/>.= 
<font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T47" title="MCART_1:th.47">MCART_1:47</a></i>
;<br/>
<b>thus </b><font color="Maroon" title="c7">o</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font> = 
<font color="Maroon" title="c7">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
<b>by </b><i><a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<br/>.= 
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c10">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_2.html#K4" title="ALTCAT_2:func.4">the_hom_sets_of</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c9">j</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="altcat_2.html#E3:24_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="altcat_2.html#E10:24_1_1"><i><font color="Green" title="E11">A14</font></i></a>, <a class="txt" href="altcat_2.html#E11:24_1_1"><i><font color="Green" title="E12">A15</font></i></a>, <a class="txt" href="altcat_2.html#E12:24_1_1"><i><font color="Green" title="E13">A16</font></i></a></i>
;<br/>


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