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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="oppcat_1.html#K2" target="_self">opp</a> </span>)</span>;<br/>



<b>assume </b><a NAME="E1:14"/><i><font color="Green">E10</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:14"/><i><font color="Green">E11</font></i>: 
(  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<a NAME="E3:14"/><b>then </b><i><font color="Green">E12</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="oppcat_1.html#E1:14"><i><font color="Green">E10</font></i></a>, <a class="ref" href="cat_1.html#T40">CAT_1:40</a></i>;<br/>
<b>thus </b> <a href="oppcat_1.html#K6" target="_self">opp</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="oppcat_1.html#K5" target="_self">opp</a> 

<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font>

<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="oppcat_1.html#K2" target="_self">opp</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="oppcat_1.html#E1:14"><i><font color="Green">E10</font></i></a>, <a class="ref" href="cat_1.html#T41">CAT_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="oppcat_1.html#K1" target="_self">~</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="oppcat_1.html#E3:14"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="oppcat_1.html#E1:14"><i><font color="Green">E10</font></i></a>, <a class="txt" href="oppcat_1.html#E2:14"><i><font color="Green">E11</font></i></a>, <a class="ref" href="cat_1.html#T41">CAT_1:41</a></i>
;<br/>


</div>
