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

<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a></span>)</span><br/>  for <font color="Olive" title="b3">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a></span>)</span> holds <br/> ( <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) iff  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Olive" title="b2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Olive" title="b1">a</font></span>)</span>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) iff  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c3">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c2">a</font></span>)</span>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) iff  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c3">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c2">a</font></span>)</span>) )</span><br/>





<a NAME="E1:9"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) iff (  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T1" title="CAT_1:th.1">CAT_1:1</a></span>;<br/>
<a NAME="E2:9"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:9"/>
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) iff  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c3">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c2">a</font></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="ref" href="cat_1.html#T1" title="CAT_1:th.1">CAT_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
