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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_4.html#NM2" title="CAT_4:NM.2">Cocartesian_category</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">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="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">c</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="c4">b</font>,<font color="Maroon" title="c3">c</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:96"/><i><font color="Green" title="E56">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">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="E2:96"/><i><font color="Green" title="E57">A2</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">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="E3:96"/><i><font color="Green" title="E58">A3</font></i>: 
( <font color="Maroon" title="c5">f</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a> or <font color="Maroon" title="c6">g</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a> )
 ;<br/>

<div><i><font color="Green" title="E59">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:96_1"/><i><font color="Green" title="E60">A5</font></i>: 
<font color="Maroon" title="c5">f</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a>
 ;<br/><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">f1</font>, <font color="Maroon" title="c9">f2</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c7">d</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:96_1"/><i><font color="Green" title="E61">A6</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">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="E3:96_1"/><i><font color="Green" title="E62">A7</font></i>: 
<font color="Maroon" title="c8">f1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span>
 ;<br/><a NAME="E4:96_1"/>
( <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c6">g</font></span>)</span></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> &amp; <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c6">g</font></span>)</span></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> &amp;  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">a</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>  &amp;  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">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_4.html#E1:96"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="cat_4.html#E2:96"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="cat_4.html#E2:96_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="cat_4.html#T72" target="_self" title="CAT_4:th.72">Th72</a>, <a class="ref" href="cat_1.html#T52" title="CAT_1:th.52">CAT_1:52</a></i>;<br/><a NAME="E5:96_1"/><b>then </b>
<font color="Maroon" title="c8">f1</font> <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">f2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="cat_4.html#E3:96_1"><i><font color="Green" title="E62">A7</font></i></a>, <a class="ref" href="cat_4.html#T73" target="_self" title="CAT_4:th.73">Th73</a></i>;<br/><b>hence </b><a NAME="E6:96_1"/>
<font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font>
 <b>by </b><i><a class="txt" href="cat_4.html#E1:96"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="cat_4.html#E1:96_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="cat_4.html#E2:96_1"><i><font color="Green" title="E61">A6</font></i></a>, <a class="ref" href="cat_1.html#T65" title="CAT_1:th.65">CAT_1:65</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:96"/><i><font color="Green" title="E60">A8</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_4.html#K20" title="CAT_4:func.20">+</a> <font color="Maroon" title="c4">b</font></span>)</span>,<font color="Maroon" title="c3">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_4.html#E1:96"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="cat_4.html#E2:96"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="cat_4.html#T70" target="_self" title="CAT_4:th.70">Th70</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:96_2"/><i><font color="Green" title="E61">A9</font></i>: 
<font color="Maroon" title="c6">g</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a>
 ;<br/><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">f1</font>, <font color="Maroon" title="c9">f2</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c7">d</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:96_2"/><i><font color="Green" title="E62">A10</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c3">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="E3:96_2"/><i><font color="Green" title="E63">A11</font></i>: 
<font color="Maroon" title="c8">f1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span>
 ;<br/><a NAME="E4:96_2"/>
( <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c6">g</font></span>)</span></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> &amp; <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c6">g</font></span>)</span></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> &amp;  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c2">a</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>  &amp;  <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">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_4.html#E1:96"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="cat_4.html#E2:96"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="cat_4.html#E2:96_2"><i><font color="Green" title="E62">A10</font></i></a>, <a class="ref" href="cat_4.html#T72" target="_self" title="CAT_4:th.72">Th72</a>, <a class="ref" href="cat_1.html#T52" title="CAT_1:th.52">CAT_1:52</a></i>;<br/><a NAME="E5:96_2"/><b>then </b>
<font color="Maroon" title="c8">f1</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">f2</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_4.html#E3:96_2"><i><font color="Green" title="E63">A11</font></i></a>, <a class="ref" href="cat_4.html#T73" target="_self" title="CAT_4:th.73">Th73</a></i>;<br/><b>hence </b><a NAME="E6:96_2"/>
<font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font>
 <b>by </b><i><a class="txt" href="cat_4.html#E2:96"><i><font color="Green" title="E57">A2</font></i></a>, <a class="txt" href="cat_4.html#E1:96_2"><i><font color="Green" title="E61">A9</font></i></a>, <a class="txt" href="cat_4.html#E2:96_2"><i><font color="Green" title="E62">A10</font></i></a>, <a class="ref" href="cat_1.html#T65" title="CAT_1:th.65">CAT_1:65</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:96"/>
<span class="p1"><a href="cat_4.html#K27" title="CAT_4:func.27">[$</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_4.html#K27" title="CAT_4:func.27">$]</a></span> is <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a>
 <b>by </b><i><a class="txt" href="cat_4.html#E3:96"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="cat_4.html#E4:96"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="cat_4.html#E5:96"><i><font color="Green" title="E60">A8</font></i></a>, <a class="ref" href="cat_1.html#T65" title="CAT_1:th.65">CAT_1:65</a></i>;<br/>


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