<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">C'</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">D</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">S</font> be    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11" target="_self">[:</a><span class="default"><font color="Maroon">C</font>,<font color="Maroon">C'</font></span><a href="cat_2.html#K11" target="_self">:]</a></span>,<font color="Maroon">D</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>reconsider </b><font color="Maroon">S'</font> = <font color="Maroon">S</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>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">D</font> ;<br/>
<b>reconsider </b><font color="Maroon">T</font> = <span class="p1">(<span class="default"><a href="cat_2.html#K3" target="_self">curry'</a> <font color="Maroon">S'</font></span>)</span> <a href="cat_2.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of 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">D</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:49_1"/>
for <font color="Olive">c</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>  ex <font color="Olive">d</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font> st <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Olive">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>proof </b></a><div class="add">

<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>consider </b><font color="Maroon">d</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E2:49_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">c'</font></span><a href="cat_2.html#K12" target="_self">]</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T97">CAT_1:97</a></i>;<br/>
<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>thus </b><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> = 
<font color="Maroon">S</font> <a href="cat_2.html#K14" target="_self">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">d</font>
<b>by </b><i><a class="ref" href="cat_2.html#T3" target="_self">Th3</a>, <a class="txt" href="cat_2.html#E7:49_1"><i><font color="Green">E24</font></i></a></i>
;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:49_1"/>
for <font color="Olive">f</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> holds <br/> ( <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Olive">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> &amp; <font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Olive">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Olive">f</font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2"><b>proof </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><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
<font color="Maroon">S</font> <a href="cat_2.html#K14" target="_self">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">c'</font></span><a href="cat_2.html#K12" target="_self">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="cat_2.html#E7:49_1"><i><font color="Green">E24</font></i></a></i>
<br/>.= 
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p4">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span></span>)</span></span><a href="cat_2.html#K12" target="_self">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><font color="Maroon">f</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span></span><a href="cat_2.html#K13" target="_self">]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_2.html#E4:49_1"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="cat_2.html#K14" target="_self">.</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T98">CAT_1:98</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>

<b>thus </b><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
<font color="Maroon">S</font> <a href="cat_2.html#K14" target="_self">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">c'</font></span><a href="cat_2.html#K12" target="_self">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="cat_2.html#E7:49_1"><i><font color="Green">E24</font></i></a></i>
<br/>.= 
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p4">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span></span>)</span></span><a href="cat_2.html#K12" target="_self">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><font color="Maroon">f</font>,<span class="p4">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span></span><a href="cat_2.html#K13" target="_self">]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="cat_2.html#E4:49_1"><i><font color="Green">E21</font></i></a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="cat_2.html#K14" target="_self">.</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T98">CAT_1:98</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>, <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/><b>assume </b><a NAME="E3:49_1"/><i><font color="Green">E19</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E4:49_1"/><i><font color="Green">E21</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/><a NAME="E5:49_1"/><i><font color="Green">E23</font></i>: 
(  <a href="cat_1.html#K2">dom</a> <span class="p1"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><font color="Maroon">g</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span></span><a href="cat_2.html#K13" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span></span>)</span></span><a href="cat_2.html#K12" target="_self">]</a></span> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span></span><a href="cat_2.html#K13" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span></span>)</span></span><a href="cat_2.html#K12" target="_self">]</a></span> )
 <b>by </b><i><a class="txt" href="cat_2.html#E4:49_1"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E6:49_1"/>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c'</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#T56">CAT_1:56</a></i>;<br/><a NAME="E7:49_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<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#K9">*</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> <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> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span>
 <b>by </b><i><a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>;<br/><a NAME="E8:49_1"/>
<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#K9">*</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>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T59">CAT_1:59</a></i>;<br/><b>hence </b><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span> = 
<font color="Maroon">S</font> <a href="cat_2.html#K14" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">S</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><font color="Maroon">g</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span></span><a href="cat_2.html#K13" target="_self">]</a></span> <a href="cat_1.html#K4">*</a> <span class="p2"><a href="cat_2.html#K13" target="_self">[</a><span class="default"><font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span></span><a href="cat_2.html#K13" target="_self">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="cat_2.html#T3" target="_self">Th3</a>, <a class="ref" href="cat_2.html#T4" target="_self">Th4</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="cat_2.html#K14" target="_self">.</a> <font color="Maroon">g</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="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="cat_2.html#K14" target="_self">.</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_2.html#T3" target="_self">Th3</a>, <a class="ref" href="cat_2.html#T4" target="_self">Th4</a>, , <a class="ref" href="cat_1.html#T99">CAT_1:99</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="cat_2.html#K14" target="_self">.</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c'</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:49"/>
<span class="p1">(<span class="default"><a href="funct_5.html#K5">curry'</a> <font color="Maroon">S</font></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">c'</font></span>)</span> is    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">C</font>,<font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T96">CAT_1:96</a></i>;<br/>


</div>
