<?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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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="b3">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>  st  <a href="cat_1.html#K7" title="CAT_1:func.7">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; <font color="Olive" title="b3">f</font> is <a href="cat_1.html#V6" title="CAT_1:attr.6">invertible</a> holds <br/>( <font color="Olive" title="b3">f</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">monic</a> &amp; <font color="Olive" title="b3">f</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">epi</a> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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>  st  <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>  &amp; <font color="Olive" title="b1">f</font> is <a href="cat_1.html#V6" title="CAT_1:attr.6">invertible</a> holds <br/>( <font color="Olive" title="b1">f</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">monic</a> &amp; <font color="Olive" title="b1">f</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">epi</a> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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>  &amp; <font color="Maroon" title="c4">f</font> is <a href="cat_1.html#V6" title="CAT_1:attr.6">invertible</a> implies ( <font color="Maroon" title="c4">f</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">monic</a> &amp; <font color="Maroon" title="c4">f</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">epi</a> ) )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:63"/><i><font color="Green" title="E32">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> 
 <b>and </b><br/><a NAME="E2:63"/><i><font color="Green" title="E33">A2</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="cat_1.html#V6" title="CAT_1:attr.6">invertible</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">monic</a> &amp; <font color="Maroon" title="c4">f</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">epi</a> )</span><br/>

<a NAME="E3:63"/><i><font color="Green" title="E34">A3</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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#E1:63"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="cat_1.html#E2:63"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="cat_1.html#T70" target="_self" title="CAT_1:th.70">Th70</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">k</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="c2">a</font><b> such that </b><br/><a NAME="E5:63"/><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">b</font>
 <b>and </b><br/><a NAME="E6:63"/><i><font color="Green" title="E36">A5</font></i>: 
<font color="Maroon" title="c5">k</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>  <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:63"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="cat_1.html#E2:63"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="cat_1.html#T70" target="_self" title="CAT_1:th.70">Th70</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">a</font>  st  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">a</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="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Olive" title="b2">h</font> holds <br/><font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font></span><br/><b>let </b><font color="Maroon" title="c7">g</font>, <font color="Maroon" title="c8">h</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">a</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="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c8">h</font> implies <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:63_1"/><i><font color="Green" title="E37">A6</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">a</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="E2:63_1"/><i><font color="Green" title="E38">A7</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c8">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font></span><br/><font color="Maroon" title="c7">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c7">g</font>
<b>by </b><i><a class="txt" href="cat_1.html#E6:63"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="cat_1.html#E1:63_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="cat_1.html#T57" target="_self" title="CAT_1:th.57">Th57</a></i>
<br/>.= 
<font color="Maroon" title="c5">k</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c8">h</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_1.html#E1:63"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="cat_1.html#E3:63"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="cat_1.html#E1:63_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="cat_1.html#E2:63_1"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="cat_1.html#T54" target="_self" title="CAT_1:th.54">Th54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c8">h</font>
<b>by </b><i><a class="txt" href="cat_1.html#E1:63"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="cat_1.html#E3:63"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="cat_1.html#E6:63"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="cat_1.html#E1:63_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="cat_1.html#T54" target="_self" title="CAT_1:th.54">Th54</a></i>
;<br/><b>hence </b><a NAME="E4:63_1"/>
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:63_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="cat_1.html#T57" target="_self" title="CAT_1:th.57">Th57</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:63"/>
<font color="Maroon" title="c4">f</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">monic</a>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:63"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="cat_1.html#T60" target="_self" title="CAT_1:th.60">Th60</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">epi</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</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="c6">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="Maroon" title="c6">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#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="b2">h</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="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font></span><br/><b>let </b><font color="Maroon" title="c7">g</font>, <font color="Maroon" title="c8">h</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="c6">c</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">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="c7">g</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">h</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font> implies <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:63_2"/><i><font color="Green" title="E37">A8</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">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="E2:63_2"/><i><font color="Green" title="E38">A9</font></i>: 
<font color="Maroon" title="c7">g</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">h</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font></span><br/><font color="Maroon" title="c7">g</font> = 
<font color="Maroon" title="c7">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">k</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_1.html#E5:63"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="cat_1.html#E1:63_2"><i><font color="Green" title="E37">A8</font></i></a>, <a class="ref" href="cat_1.html#T58" target="_self" title="CAT_1:th.58">Th58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">k</font>
<b>by </b><i><a class="txt" href="cat_1.html#E1:63"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="cat_1.html#E3:63"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="cat_1.html#E1:63_2"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="cat_1.html#E2:63_2"><i><font color="Green" title="E38">A9</font></i></a>, <a class="ref" href="cat_1.html#T54" target="_self" title="CAT_1:th.54">Th54</a></i>
<br/>.= 
<font color="Maroon" title="c8">h</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c3">b</font></span>)</span>
<b>by </b><i><a class="txt" href="cat_1.html#E1:63"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="cat_1.html#E3:63"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="cat_1.html#E5:63"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="cat_1.html#E1:63_2"><i><font color="Green" title="E37">A8</font></i></a>, <a class="ref" href="cat_1.html#T54" target="_self" title="CAT_1:th.54">Th54</a></i>
;<br/><b>hence </b><a NAME="E4:63_2"/>
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:63_2"><i><font color="Green" title="E37">A8</font></i></a>, <a class="ref" href="cat_1.html#T58" target="_self" title="CAT_1:th.58">Th58</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:63"/>
<font color="Maroon" title="c4">f</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">epi</a>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:63"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="cat_1.html#T65" target="_self" title="CAT_1:th.65">Th65</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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