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

<b>let </b><font color="Maroon">o1</font> be   <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>, <font color="Maroon">o2</font> be   <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>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:24_1_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font>, <font color="Olive">k</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> holds  <font color="Maroon">o1</font> <a href="altcat_1.html#K4">.</a> <font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">k</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="Olive">j</font>,<font color="Olive">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="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E2:24_1_2"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font>, <font color="Olive">k</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font> holds  <font color="Maroon">o2</font> <a href="altcat_1.html#K4">.</a> <font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">k</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="Olive">j</font>,<font color="Olive">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="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:24_1_2_1"/>
<font color="Maroon">a</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>then 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="E3:24_1_2_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">a</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>thus </b><font color="Maroon">o1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">o1</font> <a href="altcat_1.html#K4">.</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/>.= 
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/>
<br/>.= 
<font color="Maroon">o2</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:24_1_2"/>
<font color="Maroon">o1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">o2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
