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<b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:61"/><i><font color="Green">E20</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:61"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">f</font> is <a href="cat_1.html#V5" target="_self">invertible</a>
 ;<br/>

<a NAME="E3:61"/><i><font color="Green">E22</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_1.html#T23" target="_self">Th23</a>, , <a class="ref" href="cat_1.html#T18" target="_self">Th18</a></i>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">a</font><b> such that </b><br/><a NAME="E5:61"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">f</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E6:61"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">k</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T23" target="_self">Th23</a>, , <a class="ref" href="cat_1.html#T18" target="_self">Th18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">g</font> be    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">a</font>, <font color="Maroon">h</font> be    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">a</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:61_1"/><i><font color="Green">E51</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:61_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">f</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">h</font>
 ;<br/><font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">g</font>
<b>by </b><i>, <a class="txt" href="cat_1.html#E35"><i><font color="Green">Lemma42</font></i></a>, </i>
<br/>.= 
<font color="Maroon">k</font> <a href="cat_1.html#K9" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T23" target="_self">Th23</a>, <a class="ref" href="cat_1.html#T25" target="_self">Th25</a>, <a class="txt" href="cat_1.html#E35"><i><font color="Green">Lemma42</font></i></a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="cat_1.html#T23" target="_self">Th23</a>, <a class="ref" href="cat_1.html#T25" target="_self">Th25</a>, , <a class="txt" href="cat_1.html#E35"><i><font color="Green">Lemma42</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E4:61_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="txt" href="cat_1.html#E35"><i><font color="Green">Lemma42</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:61"/>
<font color="Maroon">f</font> is <a href="cat_1.html#V3" target="_self">monic</a>
 <b>by </b><i><a class="ref" href="cat_1.html#T23" target="_self">Th23</a>, <a class="ref" href="cat_1.html#D4" target="_self">Def4</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">g</font> be    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font>, <font color="Maroon">h</font> be    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:61_2"/><i><font color="Green">E53</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:61_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">g</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font>
 ;<br/><font color="Maroon">g</font> = 
<font color="Maroon">g</font> <a href="cat_1.html#K9" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T26" target="_self">Th26</a>, <a class="txt" href="cat_1.html#E35"><i><font color="Green">Lemma42</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font></span>)</span> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="cat_1.html#T23" target="_self">Th23</a>, <a class="ref" href="cat_1.html#T25" target="_self">Th25</a>, <a class="txt" href="cat_1.html#E35"><i><font color="Green">Lemma42</font></i></a>, , </i>
<br/>.= 
<font color="Maroon">h</font> <a href="cat_1.html#K9" target="_self">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T23" target="_self">Th23</a>, <a class="ref" href="cat_1.html#T25" target="_self">Th25</a>, <a class="ref" href="cat_1.html#T26" target="_self">Th26</a>, <a class="txt" href="cat_1.html#E35"><i><font color="Green">Lemma42</font></i></a>, </i>
;<br/><b>hence </b><a NAME="E4:61_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="txt" href="cat_1.html#E35"><i><font color="Green">Lemma42</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:61"/>
<font color="Maroon">f</font> is <a href="cat_1.html#V4" target="_self">epi</a>
 <b>by </b><i><a class="ref" href="cat_1.html#T23" target="_self">Th23</a>, </i>;<br/>


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