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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>reconsider </b><font color="Maroon">T</font> =  <a href="ens_1.html#K23" target="_self">hom??</a> <font color="Maroon">C</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="ens_1.html#K2" target="_self">Maps</a> <span class="p1">(<span class="default"><a href="ens_1.html#K17" target="_self">Hom</a> <font color="Maroon">C</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>now </b></a><div class="add"><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>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">T</font></span>)</span> <a href="cat_2.html#K1">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> = 
<font color="Maroon">T</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="cat_2.html#T3">CAT_2:3</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" target="_self">hom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">f</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" target="_self">hom</a> <font color="Maroon">a</font>,<font color="Maroon">f</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" target="_self">hom</a> <font color="Maroon">a</font>,<font color="Maroon">f</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K18" target="_self">hom</a> <font color="Maroon">a</font>,<font color="Maroon">f</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K20" target="_self">hom?-</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="ens_1.html#T10" target="_self">Th10</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:83"/>
 <a href="ens_1.html#K20" target="_self">hom?-</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K3">curry</a> <span class="p2">(<span class="default"><a href="ens_1.html#K23" target="_self">hom??</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2"><b>now </b></a><div class="add"><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>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K3">curry'</a> <font color="Maroon">T</font></span>)</span> <a href="cat_2.html#K1">.</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> = 
<font color="Maroon">T</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_2.html#T4">CAT_2:4</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K22" target="_self">hom</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K19" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="ens_1.html#T17" target="_self">Th17</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p5">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K19" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K6">Hom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><a href="ens_1.html#K19" target="_self">hom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ens_1.html#K21" target="_self">hom-?</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="ens_1.html#T11" target="_self">Th11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:83"/>
 <a href="ens_1.html#K21" target="_self">hom-?</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K5">curry'</a> <span class="p2">(<span class="default"><a href="ens_1.html#K23" target="_self">hom??</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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