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

<b>let </b><font color="Maroon">A</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">a1</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>, <font color="Maroon">a2</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">b1</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">B</font>, <font color="Maroon">b2</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">B</font>;<br/>







<b>assume </b><a NAME="E1:16"/><i><font color="Green">E16</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span>,<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:16"/><b>then </b><i><font color="Green">E17</font></i>: 
(  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">B</font>;<br/>



<b>thus </b><a NAME="E3:16"/>
( <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_2.html#K13">]</a></span> is    <a href="cat_1.html#M1">Morphism</a> of <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span>,<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span> implies ( <font color="Maroon">f</font> is    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">a1</font>,<font color="Maroon">a2</font> &amp; <font color="Maroon">g</font> is    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">b1</font>,<font color="Maroon">b2</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1"/>
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_2.html#K13">]</a></span> is    <a href="cat_1.html#M1">Morphism</a> of <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span>,<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span>
 ;<br/>

<a NAME="E2:16_1"/><b>then </b>
( <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span>,<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span> &amp;  <a href="cat_1.html#K6">Hom</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span>,<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i>, <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a>, <a class="ref" href="cat_2.html#T42">CAT_2:42</a></i>;<br/>
<a NAME="E3:16_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E4:16_1"/>
( <font color="Maroon">f</font> is    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">a1</font>,<font color="Maroon">a2</font> &amp; <font color="Maroon">g</font> is    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">b1</font>,<font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/>


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

<b>thus </b><a NAME="E4:16"/>
( <font color="Maroon">f</font> is    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">a1</font>,<font color="Maroon">a2</font> &amp; <font color="Maroon">g</font> is    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">b1</font>,<font color="Maroon">b2</font> implies <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_2.html#K13">]</a></span> is    <a href="cat_1.html#M1">Morphism</a> of <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span>,<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span> )
 <b>by </b><i><a class="txt" href="isocat_2.html#E1:16"><i><font color="Green">E16</font></i></a>, <a class="ref" href="cat_2.html#T43">CAT_2:43</a></i>;<br/>


</div>
