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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>;<br/><b>let </b><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>;<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">b</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="c3">b</font>,<font color="Maroon" title="c4">c</font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:65"/><i><font color="Green" title="E38">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">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="E2:65"/><i><font color="Green" title="E39">A2</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">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> 
 <b>and </b><br/><a NAME="E3:65"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c5">f</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>
 <b>and </b><br/><a NAME="E4:65"/><i><font color="Green" title="E41">A4</font></i>: 
<font color="Maroon" title="c6">g</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:65_1"/>
 <a href="cat_1.html#K6" title="CAT_1:func.6">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> 
 <b>by </b><i><a class="txt" href="cat_1.html#E1:65"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="cat_1.html#E2:65"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="cat_1.html#T51" target="_self" title="CAT_1:th.51">Th51</a></i>;<br/><b>thus </b><a NAME="E2:65_1"/>
<font color="Maroon" title="c6">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">f</font> is <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:65"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="cat_1.html#E2:65"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="cat_1.html#E3:65"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="cat_1.html#E4:65"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="cat_1.html#T75" target="_self" title="CAT_1:th.75">Th75</a></i>;<br/><a NAME="E3:65_1"/><i><font color="Green" title="E42">A5</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">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#E1:65"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="cat_1.html#E3:65"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="cat_1.html#T70" target="_self" title="CAT_1:th.70">Th70</a></i>;<br/><a NAME="E4:65_1"/><i><font color="Green" title="E43">A6</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">c</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>by </b><i><a class="txt" href="cat_1.html#E2:65"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="cat_1.html#E4:65"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="cat_1.html#T70" target="_self" title="CAT_1:th.70">Th70</a></i>;<br/><a NAME="E5:65_1"/><b>then </b><i><font color="Green" title="E44">A7</font></i>: 
 <a href="cat_1.html#K6" title="CAT_1:func.6">Hom</a> <font color="Maroon" title="c4">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>by </b><i><a class="txt" href="cat_1.html#E3:65_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="cat_1.html#T51" target="_self" title="CAT_1:th.51">Th51</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span></span>)</span> = 
<font color="Maroon" title="c6">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_1.html#E1:65"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="cat_1.html#E2:65"><i><font color="Green" title="E39">A2</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="c6">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_1.html#E1:65"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="cat_1.html#E3:65_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="cat_1.html#E4:65_1"><i><font color="Green" title="E43">A6</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="c6">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<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> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_1.html#E1:65"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="cat_1.html#E3:65"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="cat_1.html#D14" target="_self" title="CAT_1:def.14">Def14</a></i>
<br/>.= 
<font color="Maroon" title="c6">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="cat_1.html#E4:65_1"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="cat_1.html#T57" target="_self" title="CAT_1:th.57">Th57</a></i>
<br/>.= 
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="txt" href="cat_1.html#E2:65"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="cat_1.html#E4:65"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="cat_1.html#D14" target="_self" title="CAT_1:def.14">Def14</a></i>
;<br/><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">f</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c6">g</font></span>)</span> <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_1.html#E1:65"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="cat_1.html#E2:65"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="cat_1.html#E5:65_1"><i><font color="Green" title="E44">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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c6">g</font></span>)</span></span>)</span> <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_1.html#E2:65"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="cat_1.html#E3:65_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="cat_1.html#E4:65_1"><i><font color="Green" title="E43">A6</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p2">(<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></span>)</span> <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_1.html#E2:65"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="cat_1.html#E4:65"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="cat_1.html#D14" target="_self" title="CAT_1:def.14">Def14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span> <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_1.html#E3:65_1"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="cat_1.html#T58" target="_self" title="CAT_1:th.58">Th58</a></i>
<br/>.= 
 <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#E1:65"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="cat_1.html#E3:65"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="cat_1.html#D14" target="_self" title="CAT_1:def.14">Def14</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:65"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K11" title="CAT_1:func.11">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="cat_1.html#D14" target="_self" title="CAT_1:def.14">Def14</a></i>;<br/>


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