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

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/>







<b>assume </b><a NAME="E1:55"/><i><font color="Green" title="E29">A1</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>thus </b><a NAME="E2:55"/>
( <font color="Maroon" title="c4">f</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">epi</a> implies for <font color="Olive" title="b1">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">g1</font>, <font color="Olive" title="b3">g2</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">c</font>  st  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">g1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> holds <br/><font color="Olive" title="b2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1"/><i><font color="Green" title="E30">A2</font></i>: 
for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive" title="b2">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Olive" title="b1">g1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font> holds <br/><font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D11" target="_self" title="CAT_1:def.11">CAT_1:def 11</a><br/>

<b>let </b><font color="Maroon" title="c5">c</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c7">g2</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">c</font>;<br/>



<b>assume </b><a NAME="E2:55_1"/>
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E3:55_1"/><b>then </b>
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c7">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c7">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c6">g1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c7">g2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font> )
 
<b>by </b><i><a class="txt" href="cat_1.html#E1:55"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="cat_1.html#D13" target="_self" title="CAT_1:def.13">Def13</a>, <a class="ref" href="cat_1.html#T23" target="_self" title="CAT_1:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E4:55_1"/>
(  not <font color="Maroon" title="c6">g1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> or <font color="Maroon" title="c6">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g2</font> )
 <b>by </b><i><a class="txt" href="cat_1.html#E1:55_1"><i><font color="Green" title="E30">A2</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E3:55"/><i><font color="Green" title="E30">A3</font></i>: 
for <font color="Olive" title="b1">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">g1</font>, <font color="Olive" title="b3">g2</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">c</font>  st  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">g1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> holds <br/><font color="Olive" title="b2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">g1</font>, <font color="Maroon" title="c6">g2</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D11" target="_self" title="CAT_1:def.11">CAT_1:def 11</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E4:55"/><i><font color="Green" title="E31">A4</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E5:55"/><i><font color="Green" title="E32">A5</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c6">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E6:55"/><i><font color="Green" title="E33">A6</font></i>: 
 <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c6">g2</font>
 ;<br/>

<a NAME="E7:55"/><i><font color="Green" title="E34">A7</font></i>: 
 <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="cat_1.html#E1:55"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="cat_1.html#T23" target="_self" title="CAT_1:th.23">Th23</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">c</font> =  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">g1</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">g1</font> = <font color="Maroon" title="c5">g1</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>, <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">g1</font> <b>by </b><i><a class="txt" href="cat_1.html#E4:55"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="cat_1.html#E7:55"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="cat_1.html#T22" target="_self" title="CAT_1:th.22">Th22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">g2</font> = <font color="Maroon" title="c6">g2</font> as    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>, <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">g1</font> <b>by </b><i><a class="txt" href="cat_1.html#E5:55"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="cat_1.html#E6:55"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="cat_1.html#E7:55"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="cat_1.html#T22" target="_self" title="CAT_1:th.22">Th22</a></i>;<br/>
<a NAME="E10:55"/><i><font color="Green" title="E35">A8</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c3">b</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">g1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="cat_1.html#E4:55"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="cat_1.html#E7:55"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="cat_1.html#T18" target="_self" title="CAT_1:th.18">Th18</a></i>;<br/>
<a NAME="E11:55"/><b>then </b>
( <font color="Maroon" title="c8">g1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c9">g2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font> )
 
<b>by </b><i><a class="txt" href="cat_1.html#E1:55"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="cat_1.html#D13" target="_self" title="CAT_1:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E12:55"/>
(  not <font color="Maroon" title="c5">g1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font> or <font color="Maroon" title="c5">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g2</font> )
 <b>by </b><i><a class="txt" href="cat_1.html#E3:55"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="cat_1.html#E10:55"><i><font color="Green" title="E35">A8</font></i></a></i>;<br/>


</div>
