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<b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<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>thus </b><a NAME="E1:17"/>
( <font color="Maroon">f</font> <a href="oppcat_1.html#K5" target="_self">opp</a>  is <a href="cat_1.html#V4">epi</a> implies <font color="Maroon">f</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">E23</font></i>: 
for <font color="Olive">g1</font>, <font color="Olive">g2</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" target="_self">opp</a> </span>)</span>  st  <a href="cat_1.html#K2">dom</a> <font color="Olive">g1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">f</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">g2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">f</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">g1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">g2</font> &amp; <font color="Olive">g1</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">g2</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> holds <br/><font color="Olive">g1</font> <a href="hidden.html#R1">=</a> <font color="Olive">g2</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">f1</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>, <font color="Maroon">f2</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</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">E25</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f2</font>
 <b>and </b><br/><a NAME="E3:17_1"/><i><font color="Green">E26</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:17_1"/><i><font color="Green">E28</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:17_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f2</font>
 ;<br/>

<b>set </b><font color="Maroon">g1</font> = <font color="Maroon">f1</font> <a href="oppcat_1.html#K5" target="_self">opp</a> ;<br/>
<b>set </b><font color="Maroon">g2</font> = <font color="Maroon">f2</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 NAME="E1:17_1_1"/>
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E2:17_1_1"/>
 <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:17_1_1"/>
 <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f1</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">f</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f2</font></span>)</span> <a href="oppcat_1.html#K5" target="_self">opp</a> 
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</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">f</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<a NAME="E7:17_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="oppcat_1.html#K5" target="_self">opp</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="oppcat_1.html#K5" target="_self">opp</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:17_1"/>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 ;<br/>


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

<b>assume </b><a NAME="E2:17"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">f1</font>, <font color="Olive">f2</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st  <a href="cat_1.html#K2">dom</a> <font color="Olive">f1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Olive">f2</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">f1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">f2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Olive">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <font color="Olive">f2</font> holds <br/><font color="Olive">f1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f2</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">g1</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" target="_self">opp</a> </span>)</span>, <font color="Maroon">g2</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" 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">E45</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">f</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">E46</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">f</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">E47</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">g2</font>
 <b>and </b><br/><a NAME="E6:17"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">g1</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">f1</font> =  <a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">g1</font>;<br/>
<b>set </b><font color="Maroon">f2</font> =  <a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">g2</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 NAME="E1:17_2"/>
 <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">g1</font></span>)</span> <a href="hidden.html#R1">=</a>  <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">g2</font></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E2:17_2"/><i><font color="Green">E49</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">g1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:17_2"/><i><font color="Green">E50</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">g2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">f</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">g1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</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">g1</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">g1</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">f</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>
<b>by </b><i>, </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">g2</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">f</font> <a href="oppcat_1.html#K5" target="_self">opp</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</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">g2</font></span>)</span></span>)</span> <a href="oppcat_1.html#K5" target="_self">opp</a> 
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f</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">g2</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">g1</font> <a href="hidden.html#R1">=</a>  <a href="oppcat_1.html#K6" target="_self">opp</a> <font color="Maroon">g2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:17"/>
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font>
 ;<br/>


</div>
