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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">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>now </b></a><div class="add"><a NAME="E1:80_1"/><i><font color="Green">E16</font></i>: 
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:80_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K22" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1"><b>now </b></a><div class="add"><a NAME="E1:80_1_1"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span> <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#T19">CAT_1:19</a></i>;<br/><b>hence </b><a NAME="E2:80_1_1"/>
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="cat_1.html#T52">CAT_1:52</a></i>;<br/><a NAME="E3:80_1_1"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span> <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#T19">CAT_1:19</a></i>;<br/><b>hence </b><a NAME="E4:80_1_1"/>
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="cat_1.html#T52">CAT_1:52</a></i>;<br/></div><b>end;</b></div><a NAME="E4:80_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="ens_1.html#K18" target="_self">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" target="_self">hom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> is   <a href="funct_2.html#NM1">Function</a> of  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span>, <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T19">FUNCT_2:19</a></i>;<br/><b>hence </b><a NAME="E5:80_1"/><i><font color="Green">E17</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18" target="_self">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19" target="_self">hom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="ens_1.html#E29"><i><font color="Green">Lemma57</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E6:80_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">x</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> ;<br/><a NAME="E8:80_1"/><i><font color="Green">E20</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> )
 <b>by </b><i><a class="ref" href="ens_1.html#T19" target="_self">Th19</a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/><a NAME="E9:80_1"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/><a NAME="E10:80_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span>
 ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="ens_1.html#K22" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="ens_1.html#T19" target="_self">Th19</a>, <a class="ref" href="ens_1.html#T15" target="_self">Th15</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="ens_1.html#T20" target="_self">Th20</a>, <a class="ref" href="cat_1.html#T43">CAT_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K18" target="_self">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="txt" href="ens_1.html#E33"><i><font color="Green">Lemma61</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K18" target="_self">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K19" target="_self">hom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i><a class="ref" href="ens_1.html#T19" target="_self">Th19</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K18" target="_self">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K19" target="_self">hom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="ens_1.html#T19" target="_self">Th19</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:80"/>
 <a href="ens_1.html#K22" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" target="_self">hom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K19" target="_self">hom</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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