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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">c</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>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>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:63"/><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="E2:63"/><i><font color="Green">E21</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> 
 ;<br/>

<b>assume </b><a NAME="E3:63"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">f</font> is <a href="cat_1.html#V5" target="_self">invertible</a>
 ;<br/>

<a NAME="E4:63"/><b>then </b><i><font color="Green">E23</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#T18" target="_self">Th18</a></i>;<br/>
<b>consider </b><font color="Maroon">f1</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="E6:63"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f1</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="E7:63"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">f1</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#T26" target="_self">Th26</a>, <a class="ref" href="cat_1.html#T18" target="_self">Th18</a></i>;<br/>
<b>assume </b><a NAME="E8:63"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">g</font> is <a href="cat_1.html#V5" target="_self">invertible</a>
 ;<br/>

<a NAME="E9:63"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">b</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#T25" target="_self">Th25</a>, <a class="ref" href="cat_1.html#T18" target="_self">Th18</a></i>;<br/>
<b>consider </b><font color="Maroon">g1</font> being    <a href="cat_1.html#M1" target="_self">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">b</font><b> such that </b><br/><a NAME="E11:63"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">g</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E12:63"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">g1</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T25" target="_self">Th25</a>, , <a class="ref" href="cat_1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E13:63"/><i><font color="Green">E56</font></i>: 
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">c</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#T25" target="_self">Th25</a>, <a class="ref" href="cat_1.html#T8" target="_self">Th8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:63_1"/><i><font color="Green">E57</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>by </b><i>, , <a class="ref" href="cat_1.html#T8" target="_self">Th8</a></i>;<br/><b>take </b><font color="Maroon">f1g1</font> = <font color="Maroon">f1</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">g1</font>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">g</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">f1g1</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> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">g1</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T25" target="_self">Th25</a>, <a class="ref" href="cat_1.html#D8" target="_self">Def8</a>, </i>
<br/>.= 
<font color="Maroon">g</font> <a href="cat_1.html#K9" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">g1</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="cat_1.html#E35"><i><font color="Green">Lemma42</font></i></a>, , </i>
<br/>.= 
 <a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">c</font>
<b>by </b><i>, , </i>
;<br/><a NAME="E3:63_1"/>
 <a href="cat_1.html#K6" target="_self">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">c</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#T25" target="_self">Th25</a>, <a class="ref" href="cat_1.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><font color="Maroon">f1g1</font> <a href="cat_1.html#K9" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font></span>)</span> = 
<font color="Maroon">f1</font> <a href="cat_1.html#K9" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="cat_1.html#K9" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">f1</font> <a href="cat_1.html#K9" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T25" target="_self">Th25</a>, , , </i>
<br/>.= 
 <a href="cat_1.html#K10" target="_self">id</a> <font color="Maroon">a</font>
<b>by </b><i>, , </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:63"/>
<font color="Maroon">g</font> <a href="cat_1.html#K9" target="_self">*</a> <font color="Maroon">f</font> is <a href="cat_1.html#V5" target="_self">invertible</a>
 <b>by </b><i>, <a class="ref" href="cat_1.html#T18" target="_self">Th18</a></i>;<br/>


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