<?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> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:17"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a>  is <a href="cat_1.html#V4">epi</a> implies <font color="Maroon">c<sub>2</sub></font> is <a href="cat_1.html#V3">monic</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <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> st  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D11">CAT_1:def 11</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D10">CAT_1:def 10</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:17_1"/><i><font color="Green">E11</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#K2">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E3:17_1"/><i><font color="Green">E12</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:17_1"/><i><font color="Green">E13</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E5:17_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>now </b></a><div class="add"><b>thus </b> <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> = 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="oppcat_1.html#E3:17_1"><i><font color="Green">E12</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>

;<br/><b>thus </b> <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> = 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="oppcat_1.html#E4:17_1"><i><font color="Green">E13</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>

;<br/><b>thus </b> <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> = 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="oppcat_1.html#E2:17_1"><i><font color="Green">E11</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>

;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="oppcat_1.html#K5" target="_self">opp</a> 
<b>by </b><i><a class="txt" href="oppcat_1.html#E3:17_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="oppcat_1.html#E5:17_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="oppcat_1.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>
<b>by </b><i><a class="txt" href="oppcat_1.html#E4:17_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="oppcat_1.html#T17" target="_self">Th17</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E7:17_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> 
 
<b>by </b><i><a class="txt" href="oppcat_1.html#E1:17_1"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> 

<br/>.= 
<font color="Maroon">c<sub>4</sub></font>

;<br/><br/>


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

<b>assume </b><a NAME="E2:17"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D10">CAT_1:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D11">CAT_1:def 11</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E3:17"/><i><font color="Green">E11</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>
 <b>and </b><br/><a NAME="E4:17"/><i><font color="Green">E12</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>
 <b>and </b><br/><a NAME="E5:17"/><i><font color="Green">E13</font></i>: 
 <a href="cat_1.html#K3">cod</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>4</sub></font>
 <b>and </b><br/><a NAME="E6:17"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>thus </b> <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>3</sub></font></span>)</span> = 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="oppcat_1.html#E5:17"><i><font color="Green">E13</font></i></a></i>
<br/>.= 
 <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>4</sub></font></span>)</span>

;<br/><b>thus </b><i><font color="Green">E15</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="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>3</sub></font>

<br/>.= 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="oppcat_1.html#E3:17"><i><font color="Green">E11</font></i></a></i>
;<br/><b>thus </b><i><font color="Green">E16</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>4</sub></font></span>)</span> = 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>4</sub></font>

<br/>.= 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="oppcat_1.html#E4:17"><i><font color="Green">E12</font></i></a></i>
;<br/><b>thus </b><font color="Maroon">c<sub>2</sub></font> <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> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <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>)</span> <a href="oppcat_1.html#K5" target="_self">opp</a> 

<br/>.= 
<span class="p1">(<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> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>
<b>by </b><i><a class="txt" href="oppcat_1.html#E2:17_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="oppcat_1.html#T17" target="_self">Th17</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>
<b>by </b><i><a class="txt" href="oppcat_1.html#E6:17"><i><font color="Green">E14</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="oppcat_1.html#K5" target="_self">opp</a> 
<b>by </b><i><a class="txt" href="oppcat_1.html#E3:17_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="oppcat_1.html#T17" target="_self">Th17</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <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>4</sub></font></span>)</span>

;<br/></div><b>end;</b></div>
<a NAME="E8:17"/><b>then </b>
 <a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="oppcat_1.html#E2:17"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>hence </b><font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="oppcat_1.html#K5" target="_self">opp</a> 

<br/>.= 
<font color="Maroon">c<sub>4</sub></font>

;<br/><br/>


</div>
