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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">c</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<br/>





<b>set </b><font color="Maroon">A</font> =  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>;<br/>
<a NAME="E1:63"/><i><font color="Green">E16</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>now </b></a><div class="add"><a NAME="E1:63_1"/>
(  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><b>hence </b><a NAME="E2:63_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K18" target="_self">hom</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 <b>by </b><i>, <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="E3:63_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
 ;<br/><b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">x</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> ;<br/><a NAME="E5:63_1"/><i><font color="Green">E19</font></i>: 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="ens_1.html#T8" target="_self">Th8</a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="ens_1.html#K18" target="_self">hom</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">g</font>
<b>by </b><i>, <a class="ref" href="ens_1.html#T8" target="_self">Th8</a>, </i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="txt" href="ens_1.html#E15"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="cat_1.html#T46">CAT_1:46</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:63"/>
 <a href="ens_1.html#K18" target="_self">hom</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>


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