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

<b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>







<b>set </b><font color="Maroon">i1</font> = <font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> ;<br/>
<b>set </b><font color="Maroon">i2</font> = <font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> ;<br/>
<b>thus </b><a NAME="E1:97"/>
( <font color="Maroon">c</font> <a href="cat_3.html#R2" target="_self">is_a_product_wrt</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> implies <font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a>  <a href="cat_3.html#R4" target="_self">is_a_coproduct_wrt</a> <font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> ,<font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:97_1"/><i><font color="Green">E38</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>and </b><br/><a NAME="E2:97_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">d</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">d</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p1</font></span>)</span> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">d</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p2</font></span>)</span> holds <br/> ex <font color="Olive">h</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> st <br/>( <font color="Olive">h</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">d</font>,<font color="Maroon">c</font> &amp; ( for <font color="Olive">k</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">d</font>,<font color="Maroon">c</font> holds <br/>( ( <font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp; <font color="Maroon">p2</font> <a href="cat_1.html#K4">*</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> ) iff <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> ) ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D16" target="_self">CAT_3:def 16</a><br/>

<a NAME="E3:97_1"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a>  &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a>  )
 
<b>by </b><i><a class="ref" href="cat_3.html#T9" target="_self">Th9</a>, <a class="ref" href="oppcat_1.html#D2">OPPCAT_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:97_1"/><i><font color="Green">E43</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a>  &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a>  )
 <b>by </b><i><a class="ref" href="oppcat_1.html#T9">OPPCAT_1:9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D19" target="_self">CAT_3:def 19</a><br/>

<b>let </b><font color="Maroon">d</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>, <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>;<br/>



<b>assume </b><a NAME="E5:97_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span>,<font color="Maroon">d</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span>,<font color="Maroon">d</font> )
 ;<br/>

<a NAME="E6:97_1"/><b>then </b>
(  <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span></span>)</span> &amp;  <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T14">OPPCAT_1:14</a></i>;<br/>
<a NAME="E7:97_1"/><b>then </b>
(  <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <span class="p3">(<span class="default"><font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span></span>)</span> &amp;  <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <span class="p3">(<span class="default"><font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T12">OPPCAT_1:12</a></i>;<br/>
<a NAME="E8:97_1"/><b>then </b>
(  <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p1</font></span>)</span> &amp;  <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T7">OPPCAT_1:7</a></i>;<br/>
<b>then consider </b><font color="Maroon">h</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E10:97_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">d</font></span>)</span>,<font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E11:97_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">k</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">d</font></span>)</span>,<font color="Maroon">c</font> holds <br/>( ( <font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">f</font> &amp; <font color="Maroon">p2</font> <a href="cat_1.html#K4">*</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">g</font> ) iff <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> )
 <b>by </b><i><a class="ref" href="cat_3.html#T10" target="_self">Th10</a></i>;<br/>
<b>take </b>
<font color="Maroon">h</font> <a href="oppcat_1.html#K5">opp</a> 
;<br/>

<a NAME="E12:97_1"/>
<font color="Maroon">h</font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">d</font></span>)</span> <a href="oppcat_1.html#K3">opp</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="oppcat_1.html#T13">OPPCAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E13:97_1"/>
<font color="Maroon">h</font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="oppcat_1.html#T5">OPPCAT_1:5</a></i>;<br/>

<b>let </b><font color="Maroon">k</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>;<br/>

<b>assume </b><a NAME="E14:97_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>,<font color="Maroon">d</font>
 ;<br/>

<a NAME="E15:97_1"/><b>then </b>
 <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T14">OPPCAT_1:14</a></i>;<br/>
<a NAME="E16:97_1"/><b>then </b>
 <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4">opp</a> <font color="Maroon">d</font></span>)</span>,<font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T4">OPPCAT_1:4</a></i>;<br/>
<a NAME="E17:97_1"/><b>then </b>
( ( <font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">f</font> &amp; <font color="Maroon">p2</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">g</font> ) iff <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="txt" href="cat_3.html#E10:97_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E18:97_1"/><b>then </b>
( ( <font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="oppcat_1.html#K5">opp</a>  &amp; <font color="Maroon">p2</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="oppcat_1.html#K5">opp</a>  ) iff <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="oppcat_1.html#K5">opp</a>  )
 
<b>by </b><i><a class="ref" href="oppcat_1.html#D5">OPPCAT_1:def 5</a></i>;<br/>
<a NAME="E19:97_1"/><b>then </b>
( ( <font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">p2</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> ) iff <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="oppcat_1.html#D4">OPPCAT_1:def 4</a></i>;<br/>
<a NAME="E20:97_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> implies <font color="Maroon">h</font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> ) &amp; ( <font color="Maroon">h</font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> implies ( <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> ) ) &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a>  )
 
<b>by </b><i><a class="txt" href="cat_3.html#E11:97_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a>, <a class="ref" href="oppcat_1.html#T7">OPPCAT_1:7</a>, <a class="ref" href="oppcat_1.html#D4">OPPCAT_1:def 4</a></i>;<br/>
<a NAME="E21:97_1"/><b>then </b>
( (  <a href="oppcat_1.html#K6">opp</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="oppcat_1.html#K6">opp</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> ) iff <font color="Maroon">h</font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, <a class="ref" href="oppcat_1.html#T19">OPPCAT_1:19</a></i>;<br/>
<a NAME="E22:97_1"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> ) iff <font color="Maroon">h</font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 
<b>by </b><i><a class="ref" href="oppcat_1.html#D5">OPPCAT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E23:97_1"/>
( ( <font color="Maroon">k</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">k</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> ) iff <font color="Maroon">h</font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="oppcat_1.html#D4">OPPCAT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E2:97"/><i><font color="Green">E78</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a>  &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a>  )
 <b>and </b><br/><a NAME="E3:97"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">d</font> being  <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span><br/> for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span>,<font color="Olive">d</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span>,<font color="Olive">d</font> holds <br/> ex <font color="Olive">h</font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span> st <br/>( <font color="Olive">h</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>,<font color="Olive">d</font> &amp; ( for <font color="Olive">k</font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>,<font color="Olive">d</font> holds <br/>( ( <font color="Olive">k</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp; <font color="Olive">k</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> ) iff <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> ) ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D19" target="_self">CAT_3:def 19</a><br/>

<a NAME="E4:97"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a>  &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a>  )
 
<b>by </b><i><a class="ref" href="cat_3.html#T9" target="_self">Th9</a>, <a class="ref" href="oppcat_1.html#T9">OPPCAT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E5:97"/><i><font color="Green">E97</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="oppcat_1.html#D2">OPPCAT_1:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_3.html#D16" target="_self">CAT_3:def 16</a><br/>

<b>let </b><font color="Maroon">d</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>





<b>assume </b><a NAME="E6:97"/>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p1</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p2</font></span>)</span> )
 ;<br/>

<a NAME="E7:97"/><b>then </b>
( <font color="Maroon">f</font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p1</font></span>)</span> <a href="oppcat_1.html#K3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span> &amp; <font color="Maroon">g</font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">p2</font></span>)</span> <a href="oppcat_1.html#K3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T13">OPPCAT_1:13</a></i>;<br/>
<a NAME="E8:97"/><b>then </b>
( <font color="Maroon">f</font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span> &amp; <font color="Maroon">g</font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T11">OPPCAT_1:11</a></i>;<br/>
<b>then consider </b><font color="Maroon">h</font> being   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span><b> such that </b><br/><a NAME="E10:97"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>
 <b>and </b><br/><a NAME="E11:97"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">k</font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="oppcat_1.html#K2">opp</a> </span>)</span>  st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span> holds <br/>( ( <font color="Olive">k</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="oppcat_1.html#K5">opp</a>  &amp; <font color="Olive">k</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="oppcat_1.html#K5">opp</a>  ) iff <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> )
 <b>by </b><i><a class="ref" href="cat_3.html#T10" target="_self">Th10</a></i>;<br/>
<b>take </b>
 <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">h</font>
;<br/>

<a NAME="E12:97"/>
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6">opp</a> <font color="Maroon">h</font></span>)</span> <a href="oppcat_1.html#K5">opp</a> 
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T8">OPPCAT_1:8</a></i>;<br/>
<b>hence </b><a NAME="E13:97"/>
 <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="oppcat_1.html#T13">OPPCAT_1:13</a></i>;<br/>

<b>let </b><font color="Maroon">k</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E14:97"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">d</font>,<font color="Maroon">c</font>
 ;<br/>

<a NAME="E15:97"/><b>then </b>
<font color="Maroon">k</font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="oppcat_1.html#K3">opp</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T13">OPPCAT_1:13</a></i>;<br/>
<a NAME="E16:97"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="oppcat_1.html#K5">opp</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="oppcat_1.html#K5">opp</a>  ) iff <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="oppcat_1.html#K5">opp</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E17:97"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> implies <font color="Maroon">h</font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="oppcat_1.html#K5">opp</a>  ) &amp; ( <font color="Maroon">h</font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="oppcat_1.html#K5">opp</a>  implies ( <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="oppcat_1.html#K5">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> ) ) &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a>, <a class="ref" href="oppcat_1.html#D4">OPPCAT_1:def 4</a></i>;<br/>
<a NAME="E18:97"/><b>then </b>
( ( <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font></span>)</span> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font></span>)</span> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> ) iff <font color="Maroon">h</font> <a href="oppcat_1.html#K5">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, <a class="ref" href="oppcat_1.html#T17">OPPCAT_1:17</a>, <a class="ref" href="oppcat_1.html#D4">OPPCAT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E19:97"/>
( ( <font color="Maroon">p1</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">p2</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> ) iff  <a href="oppcat_1.html#K6">opp</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="oppcat_1.html#D4">OPPCAT_1:def 4</a>, <a class="ref" href="oppcat_1.html#D5">OPPCAT_1:def 5</a></i>;<br/>


</div>
