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

<b>let </b><font color="Maroon">A</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">B</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">F1</font> be    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <font color="Maroon">F2</font> be    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>;<br/>







<b>assume </b><a NAME="E1:17"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 ;<br/>

<a NAME="E2:17"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">F2</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/>
<b>let </b><font color="Maroon">u</font> be    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/>



<a NAME="E3:17"/>
the <a href="cat_1.html#U1">Objects</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="cat_2.html#T33">CAT_2:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">u</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#U1">Objects</a> of <font color="Maroon">A</font>,the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> ;<br/>
<a NAME="E5:17"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">B</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a><br/>

<a NAME="E1:17_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="cat_2.html#K12">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="cat_2.html#K12">]</a></span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:17_1"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</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="txt" href="isocat_2.html#E2:17"><i><font color="Green">E17</font></i></a>, <a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">B</font>;<br/><i><font color="Green">E50</font></i>: <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> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> = 
<font color="Maroon">t</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="cat_2.html#T3">CAT_2:3</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="nattra_1.html#K5">.</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="cat_2.html#K12">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E2:17"><i><font color="Green">E17</font></i></a>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
;<br/><a NAME="E2:17_2"/><i><font color="Green">E51</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="cat_2.html#K12">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="cat_2.html#K12">]</a></span> )
 <b>by </b><i/>;<br/><a NAME="E3:17_2"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</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="isocat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/><b>hence </b><a NAME="E4:17_2"/>
<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> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">B</font>;<br/><a NAME="E1:17_3"/>
<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> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="isocat_2.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E2:17_3"/>
<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> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> is    <a href="cat_1.html#M1">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">s</font> = <span class="p1">(<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> <font color="Maroon">a</font> as    <a href="nattra_1.html#M1">transformation</a> of <font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font>,<font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font> <b>by </b><i><a class="ref" href="isocat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="nattra_1.html#D3">NATTRA_1:def 3</a></i>;<br/>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b1</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">B</font>, <font color="Maroon">b2</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">B</font>;<br/><b>assume </b><a NAME="E1:17_4"/><i><font color="Green">E55</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">b1</font>,<font color="Maroon">b2</font>;<br/><a NAME="E2:17_4"/><i><font color="Green">E57</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">a</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/><b>then reconsider </b><font color="Maroon">g</font> = <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span>,<font color="Maroon">f</font></span><a href="cat_2.html#K13">]</a></span> as    <a href="cat_1.html#M1">Morphism</a> of <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span>,<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span> <b>by </b><i>, <a class="ref" href="cat_2.html#T43">CAT_2:43</a></i>;<br/><a NAME="E4:17_4"/><i><font color="Green">E58</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span>,<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></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="isocat_2.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E5:17_4"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span></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#T126">CAT_1:126</a></i>;<br/><a NAME="E6:17_4"/><i><font color="Green">E60</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span></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="txt" href="isocat_2.html#E2:17"><i><font color="Green">E17</font></i></a>, <a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/><a NAME="E7:17_4"/><i><font color="Green">E61</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</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#T126">CAT_1:126</a></i>;<br/><a NAME="E8:17_4"/><i><font color="Green">E62</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span> )
 <b>by </b><i/>;<br/><a NAME="E9:17_4"/><i><font color="Green">E65</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span></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="txt" href="isocat_2.html#E2:17"><i><font color="Green">E17</font></i></a>, <a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/><a NAME="E10:17_4"/><i><font color="Green">E66</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span></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#T126">CAT_1:126</a></i>;<br/><a NAME="E11:17_4"/><i><font color="Green">E67</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</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#T126">CAT_1:126</a></i>;<br/><a NAME="E12:17_4"/><i><font color="Green">E68</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span> )
 <b>by </b><i/>;<br/><i><font color="Green">E69</font></i>: <font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b1</font> = 
<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> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="isocat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="cat_2.html#T3">CAT_2:3</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="nattra_1.html#K5">.</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E2:17"><i><font color="Green">E17</font></i></a>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
;<br/><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font> = 
<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> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b2</font>
<b>by </b><i><a class="ref" href="isocat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b2</font>
<b>by </b><i><a class="ref" href="cat_2.html#T3">CAT_2:3</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="nattra_1.html#K5">.</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E2:17"><i><font color="Green">E17</font></i></a>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nattra_1.html#K5">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="ref" href="isocat_2.html#T4" target="_self">Th4</a>, <a class="ref" href="isocat_2.html#T5" target="_self">Th5</a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nattra_1.html#K5">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nattra_1.html#K5">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K14">.</a> <span class="p2">(<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>
<b>by </b><i><a class="ref" href="cat_2.html#T47">CAT_2:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nattra_1.html#K5">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nattra_1.html#K5">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b2</font></span><a href="cat_2.html#K12">]</a></span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nattra_1.html#K5">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_2.html#E1:17"><i><font color="Green">E16</font></i></a>, , <a class="ref" href="nattra_1.html#D8">NATTRA_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="nattra_1.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">u</font> <a href="nattra_1.html#K5">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span></span>)</span>
<b>by </b><i>, , <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K14">.</a> <span class="p2">(<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> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nattra_1.html#K5">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="nattra_1.html#K5">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b1</font></span><a href="cat_2.html#K12">]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="cat_2.html#T47">CAT_2:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i>, <a class="txt" href="isocat_2.html#E5:17"><i><font color="Green">E32</font></i></a>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:17"/>
<font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="isocat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/>

<b>hence </b><a NAME="E11:17"/>
<span class="p1">(<span class="default"><a href="funct_5.html#K3">curry</a> <font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> is    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font>,<font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#D8">NATTRA_1:def 8</a></i>;<br/>


</div>
