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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">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="b4">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/>  for <font color="Olive" title="b5">g</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">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#K3" title="CAT_1:func.3">Hom</a> (<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">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="b5">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b4">f</font> is  <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a>  holds <br/><font color="Olive" title="b5">g</font> is  <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being    <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/>  for <font color="Olive" title="b2">g</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="Maroon" title="c4">c</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">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#K3" title="CAT_1:func.3">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>  &amp; <font color="Olive" title="b2">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive" title="b1">f</font> is  <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a>  holds <br/><font color="Olive" title="b2">g</font> is  <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">g</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="Maroon" title="c4">c</font>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">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#K3" title="CAT_1:func.3">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>  &amp; <font color="Olive" title="b1">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font> is  <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a>  holds <br/><font color="Olive" title="b1">g</font> is  <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a> </span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">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#K3" title="CAT_1:func.3">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>  &amp; <font color="Maroon" title="c6">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font> is  <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a>  implies <font color="Maroon" title="c6">g</font> is  <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a>  )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">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> 
 <span class="kw">and </span><br/><a NAME="E2:48"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">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> 
 <span class="kw">and </span><br/><a NAME="E3:48"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<font color="Maroon" title="c6">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font> is  <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">g</font> is  <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">h1</font>, <font color="Olive" title="b2">h2</font> being    <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>  st  <a href="cat_1.html#K3" title="CAT_1:func.3">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>  &amp; <font color="Olive" title="b1">h1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">g</font> holds <br/><font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h2</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cat_1.html#K3" title="CAT_1:func.3">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>  &amp; <font color="Maroon" title="c8">h1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">g</font> implies <font color="Maroon" title="c8">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h2</font> )</span><br/><a NAME="E1:48_1"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="cat_1.html#T23" target="_self" title="CAT_1:th.23">Th51</a></span>;<br/><span class="kw">assume </span><a NAME="E2:48_1"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">h1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">g</font> implies <font color="Maroon" title="c8">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h2</font> )</span><br/><a NAME="E3:48_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">h1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c9">h2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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="c9">h2</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="cat_1.html#T25" target="_self" title="CAT_1:th.25">Th54</a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_1"/>
( <font color="Maroon" title="c8">h1</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</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#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c6">g</font> implies <font color="Maroon" title="c8">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E2:48_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="cat_1.html#T36" target="_self" title="CAT_1:th.36">Th65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:48"/>
<font color="Maroon" title="c6">g</font> is  <a href="cat_1.html#V4" title="CAT_1:attr.4">epi</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="cat_1.html#T36" target="_self" title="CAT_1:th.36">Th65</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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