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<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>, <font color="Maroon" title="c4">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="c5">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>let </b><font color="Maroon" title="c6">g</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="c4">c</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:53"/><i><font color="Green" title="E28">A1</font></i>: 
(  <a href="cat_1.html#K6" title="CAT_1:func.6">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>  &amp;  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">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>  )
 <b>and </b><br/><a NAME="E2:53"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c5">f</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a>
 <b>and </b><br/><a NAME="E3:53"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c6">g</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a>
 ;<br/>

<a NAME="E4:53"/><i><font color="Green" title="E31">A4</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">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> 
 
<b>by </b><i><a class="txt" href="cat_1.html#E1:53"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="cat_1.html#T51" target="_self" title="CAT_1:th.51">Th51</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">d</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="c8">h1</font>, <font color="Maroon" title="c9">h2</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">d</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:53_1"/><i><font color="Green" title="E32">A5</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">d</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> 
 <b>and </b><br/><a NAME="E2:53_1"/><i><font color="Green" title="E33">A6</font></i>: 
<font color="Maroon" title="c8">h1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">f</font></span>)</span>
 ;<br/><a NAME="E3:53_1"/>
( <font color="Maroon" title="c8">h1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">f</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">h2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">f</font></span>)</span> &amp;  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">d</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>  )
 <b>by </b><i><a class="txt" href="cat_1.html#E1:53"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="cat_1.html#E1:53_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="cat_1.html#T51" target="_self" title="CAT_1:th.51">Th51</a>, <a class="ref" href="cat_1.html#T54" target="_self" title="CAT_1:th.54">Th54</a></i>;<br/><a NAME="E4:53_1"/><b>then </b>
<font color="Maroon" title="c8">h1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c6">g</font>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:53"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="cat_1.html#E2:53"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="cat_1.html#E2:53_1"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="cat_1.html#T65" target="_self" title="CAT_1:th.65">Th65</a></i>;<br/><b>hence </b><a NAME="E5:53_1"/>
<font color="Maroon" title="c8">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h2</font>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:53"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="cat_1.html#E3:53"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="cat_1.html#E1:53_1"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="cat_1.html#T65" target="_self" title="CAT_1:th.65">Th65</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:53"/>
<font color="Maroon" title="c6">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">f</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a>
 <b>by </b><i><a class="txt" href="cat_1.html#E4:53"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="cat_1.html#T65" target="_self" title="CAT_1:th.65">Th65</a></i>;<br/>


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