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<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="E1:31"/><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="E2:31"/><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">t</font> be    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font>;<br/>

<a NAME="E3:31"/>
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">s</font> = <font color="Maroon">t</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/>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">S1</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>, <font color="Maroon">S2</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/><b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> )
 ;<br/><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>: <font color="Maroon">S1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font> = 
<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>
<b>by </b><i>, </i>
<br/>.= 
<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>
<b>by </b><i/>
;<br/><i><font color="Green">E51</font></i>: <font color="Maroon">S2</font> <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>, </i>
<br/>.= 
<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/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">s</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">s</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">t</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="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
;<br/><b>hence </b><a NAME="E5:31_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">s</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"><font color="Maroon">S1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="isocat_2.html#E5:31"><i><font color="Green">E32</font></i></a>, <a class="txt" href="isocat_2.html#E1:31_1"><i><font color="Green">E33</font></i></a>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">S1</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>, <font color="Maroon">S2</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/><b>assume </b><a NAME="E1:31_2"/>
( <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> )
 ;<br/><a NAME="E2:31_2"/><b>then </b>
for <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">B</font>  holds  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="cat_1.html#K13">.</a> <font color="Olive">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="cat_1.html#K13">.</a> <font color="Olive">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/>;<br/><b>hence </b><a NAME="E3:31_2"/>
<font color="Maroon">S1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">S2</font>
 <b>by </b><i><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">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">S1</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>, <font color="Maroon">S2</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/><b>assume </b><a NAME="E1:31_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> )
 ;<br/><b>thus </b><a NAME="E2:31_3"/>
<span class="p1">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">s</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">a</font> is    <a href="nattra_1.html#M1">transformation</a> of <font color="Maroon">S1</font>,<font color="Maroon">S2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_3_1"/>
<font color="Maroon">S1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">S2</font>
 <b>by </b><i>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D3">NATTRA_1:def 3</a><br/>

<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="E2:31_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">s</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"><font color="Maroon">S1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:31_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">s</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 <font color="Maroon">S1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font>,<font color="Maroon">S2</font> <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></div><b>end;</b></div>
<div><i><font color="Green">E58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">S1</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>, <font color="Maroon">S2</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">T</font> be    <a href="nattra_1.html#M1">transformation</a> of <font color="Maroon">S1</font>,<font color="Maroon">S2</font>;<br/><b>assume </b><a NAME="E1:31_4"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">T</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">s</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">a</font> )
 ;<br/><a NAME="E2:31_4"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">S1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">S2</font>
 <b>by </b><i/>;<br/><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="E3:31_4"/><i><font color="Green">E61</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="E4:31_4"/><i><font color="Green">E62</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="isocat_2.html#T6" target="_self">Th6</a>, <a class="ref" href="cat_2.html#T43">CAT_2:43</a></i>;<br/><a NAME="E6:31_4"/><i><font color="Green">E65</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#T6" target="_self">Th6</a>, , </i>;<br/><a NAME="E7:31_4"/><b>then </b><i><font color="Green">E66</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="E8:31_4"/><i><font color="Green">E67</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="isocat_2.html#T7" target="_self">Th7</a>, <a class="ref" href="cat_1.html#T126">CAT_1:126</a></i>;<br/><a NAME="E9:31_4"/><i><font color="Green">E68</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="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/><a NAME="E10:31_4"/><i><font color="Green">E69</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="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/><a NAME="E11:31_4"/><i><font color="Green">E70</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S1</font> <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="isocat_2.html#T6" target="_self">Th6</a>, <a class="ref" href="cat_1.html#T126">CAT_1:126</a></i>;<br/><a NAME="E12:31_4"/><i><font color="Green">E71</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S2</font> <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="isocat_2.html#T6" target="_self">Th6</a>, <a class="ref" href="cat_1.html#T126">CAT_1:126</a></i>;<br/><a NAME="E13:31_4"/><i><font color="Green">E74</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</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="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/><a NAME="E14:31_4"/><i><font color="Green">E75</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S2</font> <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="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/><i><font color="Green">E76</font></i>: <font color="Maroon">T</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b1</font> = 
<font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
<br/>.= 
<font color="Maroon">s</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">t</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="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
;<br/><i><font color="Green">E77</font></i>: <font color="Maroon">T</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font> = 
<font color="Maroon">T</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b2</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
<br/>.= 
<font color="Maroon">s</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">t</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="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
;<br/><i><font color="Green">E80</font></i>: <font color="Maroon">S1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font> = 
<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="nattra_1.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i>, </i>
<br/>.= 
<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="funct_2.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
<font color="Maroon">F1</font> <a href="cat_2.html#K14">.</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#T47">CAT_2:47</a></i>
<br/>.= 
<font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="isocat_2.html#T7" target="_self">Th7</a>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
;<br/><i><font color="Green">E81</font></i>: <font color="Maroon">S2</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</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="nattra_1.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i>, </i>
<br/>.= 
<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="funct_2.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
<font color="Maroon">F2</font> <a href="cat_2.html#K14">.</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#T47">CAT_2:47</a></i>
<br/>.= 
<font color="Maroon">F2</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="isocat_2.html#T7" target="_self">Th7</a>, <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">T</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"><font color="Maroon">S1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">f</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">t</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="isocat_2.html#T9" target="_self">Th9</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">t</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="isocat_2.html#T8" target="_self">Th8</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">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">t</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="isocat_2.html#T7" target="_self">Th7</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">S2</font> <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">T</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/>.= 
<span class="p1">(<span class="default"><font color="Maroon">S2</font> <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">T</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="isocat_2.html#D1" target="_self">Def1</a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">f</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font><br/> for <font color="Olive">s</font> being  <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>  st <font color="Maroon">t</font> <a href="funct_2.html#R1">=</a> <font color="Olive">s</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Olive">f</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Olive">f</font></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Olive">s</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Olive">f</font></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/>
<div><i><font color="Green">E82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:31_5"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">m</font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font> ;<br/><b>take </b><font color="Maroon">o</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">s</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/><b>reconsider </b><font color="Maroon">S1</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font>, <font color="Maroon">S2</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">T</font> = <span class="p1">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">s</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">S1</font>,<font color="Maroon">S2</font> <b>by </b><i><a class="txt" href="isocat_2.html#E5:31"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E5:31_5"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">S1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">S2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_5_1"/>
<font color="Maroon">S1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">S2</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a><br/>

<b>take </b>
<font color="Maroon">T</font>
;<br/>

<b>thus </b><a NAME="E2:31_5_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">B</font>  holds <br/>(  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds  <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E6:31_5"/>
<font color="Maroon">T</font> is    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">S1</font>,<font color="Maroon">S2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_5_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_5_2"/>
<font color="Maroon">S1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">S2</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D8">NATTRA_1:def 8</a><br/>

<b>thus </b><a NAME="E2:31_5_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">B</font>  holds <br/>(  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds  <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E7:31_5"/><b>then </b>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11">NatTrans</a> <font color="Maroon">B</font>,<font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#T35">NATTRA_1:35</a></i>;<br/><b>hence </b><a NAME="E8:31_5"/>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/><b>thus </b><a NAME="E9:31_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">o</font>]
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>,the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span><b> such that </b><br/><a NAME="E11:31"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">m</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<div><i><font color="Green">E93</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/><b>reconsider </b><font color="Maroon">S1</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font>, <font color="Maroon">S2</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">T</font> = <span class="p1">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">s</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">S1</font>,<font color="Maroon">S2</font> <b>by </b><i><a class="txt" href="isocat_2.html#E5:31"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E3:31_6"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">S1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">S2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_6_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_6_1"/>
<font color="Maroon">S1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">S2</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a><br/>

<b>take </b>
<font color="Maroon">T</font>
;<br/>

<b>thus </b><a NAME="E2:31_6_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">B</font>  holds <br/>(  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds  <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:31_6"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">T</font> is    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">S1</font>,<font color="Maroon">S2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_6_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_6_2"/>
<font color="Maroon">S1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">S2</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D8">NATTRA_1:def 8</a><br/>

<b>thus </b><a NAME="E2:31_6_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">B</font>  holds <br/>(  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds  <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E5:31_6"/>
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">T</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E6:31_6"/><b>then </b>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> )
 <b>by </b><i><a class="ref" href="isocat_2.html#T7" target="_self">Th7</a>, <a class="ref" href="nattra_1.html#T39">NATTRA_1:39</a></i>;<br/><b>hence </b><a NAME="E7:31_6"/>
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</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"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:31"/><b>then </b><i><font color="Green">E96</font></i>: 
for <font color="Olive">a</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>  holds  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Olive">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Olive">a</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E14:31"/><b>then </b><i><font color="Green">E97</font></i>: 
 <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font> <a href="nattra_1.html#R1">is_transformable_to</a>  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/>
<a NAME="E15:31"/>
<font color="Maroon">G</font> is    <a href="nattra_1.html#M1">transformation</a> of  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font>, <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_7"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_7"/>
 <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font> <a href="nattra_1.html#R1">is_transformable_to</a>  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D3">NATTRA_1:def 3</a><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="E2:31_7"/>
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</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"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="isocat_2.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E3:31_7"/>
<font color="Maroon">G</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> is    <a href="cat_1.html#M1">Morphism</a> of <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</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">G</font> = <font color="Maroon">G</font> as    <a href="nattra_1.html#M1">transformation</a> of  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font>, <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font> ;<br/>
<div><i><font color="Green">E98</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/><b>assume </b><a NAME="E1:31_8"/><i><font color="Green">E99</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</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">a</font>,<font color="Maroon">b</font>;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> ;<br/><a NAME="E3:31_8"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E4:31_8"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <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#R1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</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#T23">CAT_1:23</a></i>;<br/><a NAME="E5:31_8"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><a NAME="E6:31_8"/><b>then </b><i><font color="Green">E139</font></i>: 
<font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/><a NAME="E7:31_8"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><a NAME="E8:31_8"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/><a NAME="E9:31_8"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E10:31_8"/><b>then </b><i><font color="Green">E143</font></i>: 
<font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <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#R1">=</a> <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</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#T23">CAT_1:23</a></i>;<br/><b>reconsider </b><font color="Maroon">S1</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font>, <font color="Maroon">S2</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font>, <font color="Maroon">S3</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font>, <font color="Maroon">S4</font> = <span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font> <b>by </b><i/>;<br/><a NAME="E12:31_8"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">S1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">S2</font>
 <b>by </b><i>, <a class="ref" href="isocat_2.html#T9" target="_self">Th9</a>, <a class="ref" href="isocat_2.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E13:31_8"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">S1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">S2</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/><a NAME="E14:31_8"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">S2</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">S4</font>
 <b>by </b><i>, <a class="ref" href="isocat_2.html#T9" target="_self">Th9</a>, , <a class="txt" href="isocat_2.html#E2:31"><i><font color="Green">E17</font></i></a></i>;<br/><a NAME="E15:31_8"/><b>then </b><i><font color="Green">E149</font></i>: 
<font color="Maroon">S2</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">S4</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/><a NAME="E16:31_8"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">S1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">S3</font>
 <b>by </b><i>, , <a class="ref" href="isocat_2.html#D1" target="_self">Def1</a>, <a class="txt" href="isocat_2.html#E2:31"><i><font color="Green">E17</font></i></a></i>;<br/><a NAME="E17:31_8"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">S1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">S3</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/><a NAME="E18:31_8"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">S3</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">S4</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E19:31_8"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">S3</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">S4</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/><a NAME="E20:31_8"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">S1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">S4</font>
 <b>by </b><i>, , <a class="ref" href="nattra_1.html#T19">NATTRA_1:19</a></i>;<br/><b>reconsider </b><font color="Maroon">T13</font> = <font color="Maroon">F1</font> <a href="isocat_2.html#K4" target="_self">?-</a> <font color="Maroon">f</font> as    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">S1</font>,<font color="Maroon">S3</font> <b>by </b><i>, <a class="ref" href="isocat_2.html#T8" target="_self">Th8</a>, <a class="ref" href="isocat_2.html#D1" target="_self">Def1</a>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/><b>reconsider </b><font color="Maroon">T12</font> = <span class="p1">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">s</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">a</font> as    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">S1</font>,<font color="Maroon">S2</font> <b>by </b><i>, <a class="ref" href="isocat_2.html#T9" target="_self">Th9</a>, <a class="ref" href="isocat_2.html#D1" target="_self">Def1</a>, </i>;<br/><b>reconsider </b><font color="Maroon">T24</font> = <font color="Maroon">F2</font> <a href="isocat_2.html#K4" target="_self">?-</a> <font color="Maroon">f</font> as    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">S2</font>,<font color="Maroon">S4</font> <b>by </b><i>, <a class="ref" href="isocat_2.html#T9" target="_self">Th9</a>, , <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>;<br/><b>reconsider </b><font color="Maroon">T34</font> = <span class="p1">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">s</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">b</font> as    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">S3</font>,<font color="Maroon">S4</font> <b>by </b><i>, , , </i>;<br/><a NAME="E25:31_8"/><i><font color="Green">E159</font></i>: 
<span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S3</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">T13</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="isocat_2.html#T8" target="_self">Th8</a>, , <a class="ref" href="isocat_2.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E26:31_8"/><i><font color="Green">E160</font></i>: 
<span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">S2</font>,<font color="Maroon">S4</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">T24</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , <a class="ref" href="isocat_2.html#T3" target="_self">Th3</a></i>;<br/><i><font color="Green">E161</font></i>: <font color="Maroon">G</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b</font> = 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="isocat_2.html#T7" target="_self">Th7</a>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">S3</font>,<font color="Maroon">S4</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">T34</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
;<br/><i><font color="Green">E162</font></i>: <font color="Maroon">G</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="isocat_2.html#T7" target="_self">Th7</a>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">T12</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
;<br/><a NAME="E29:31_8"/><i><font color="Green">E163</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</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#T6" target="_self">Th6</a></i>;<br/><a NAME="E30:31_8"/><i><font color="Green">E164</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</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="cat_1.html#T126">CAT_1:126</a></i>;<br/><a NAME="E31:31_8"/><i><font color="Green">E165</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</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"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</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#T6" target="_self">Th6</a></i>;<br/><a NAME="E32:31_8"/><i><font color="Green">E166</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</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="cat_1.html#T126">CAT_1:126</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_8_3"><b>now </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">B</font>;<br/><a NAME="E1:31_8_3"/><i><font color="Green">E167</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S4</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c</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="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/><a NAME="E2:31_8_3"/><i><font color="Green">E168</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c</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="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/><a NAME="E3:31_8_3"/><i><font color="Green">E169</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">S3</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S4</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c</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="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/><a NAME="E4:31_8_3"/><i><font color="Green">E170</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S3</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">c</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#E6:31"><i><font color="Green">E54</font></i></a>, <a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/><a NAME="E5:31_8_3"/><i><font color="Green">E171</font></i>: 
 <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="E6:31_8_3"/><b>then </b><i><font color="Green">E172</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">c</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">b</font>,<font color="Maroon">c</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>, </i>;<br/><a NAME="E7:31_8_3"/><i><font color="Green">E173</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">c</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">c</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="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/><a NAME="E8:31_8_3"/><i><font color="Green">E174</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">c</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">b</font>,<font color="Maroon">c</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="E9:31_8_3"/><i><font color="Green">E175</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">b</font>,<font color="Maroon">c</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">b</font>,<font color="Maroon">c</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="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/><a NAME="E10:31_8_3"/><i><font color="Green">E176</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">c</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">b</font>,<font color="Maroon">c</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/><b>reconsider </b><font color="Maroon">fi</font> = <span class="p1"><a href="cat_2.html#K13">[</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">]</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">c</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">b</font>,<font color="Maroon">c</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="E12:31_8_3"/>
the <a href="cat_1.html#U2">Morphisms</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#U2">Morphisms</a> of <font color="Maroon">A</font>,the <a href="cat_1.html#U2">Morphisms</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">FF1</font> = <font color="Maroon">F1</font>, <font color="Maroon">FF2</font> = <font color="Maroon">F2</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">A</font>,the <a href="cat_1.html#U2">Morphisms</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/><i><font color="Green">E180</font></i>: <font color="Maroon">F2</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">fi</font> = 
<font color="Maroon">FF2</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">c</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"><a href="isocat_2.html#K3" target="_self">curry</a> <font color="Maroon">F2</font>,<font color="Maroon">f</font></span>)</span> <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>
<b>by </b><i><a class="ref" href="cat_2.html#T3">CAT_2:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K3" target="_self">curry</a> <font color="Maroon">F2</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">B</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#D5">CAT_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="isocat_2.html#K4" target="_self">?-</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">T24</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
;<br/><i><font color="Green">E181</font></i>: <font color="Maroon">t</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">c</font></span><a href="cat_2.html#K12">]</a></span> = 
<font color="Maroon">s</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">s</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">c</font>
<b>by </b><i><a class="ref" href="cat_2.html#T3">CAT_2:3</a></i>
<br/>.= 
<font color="Maroon">T12</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
;<br/><i><font color="Green">E182</font></i>: <font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">fi</font> = 
<font color="Maroon">FF1</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">c</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"><a href="isocat_2.html#K3" target="_self">curry</a> <font color="Maroon">F1</font>,<font color="Maroon">f</font></span>)</span> <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>
<b>by </b><i><a class="ref" href="cat_2.html#T3">CAT_2:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K3" target="_self">curry</a> <font color="Maroon">F1</font>,<font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">B</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#D5">CAT_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="isocat_2.html#K4" target="_self">?-</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">T13</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E6:31"><i><font color="Green">E54</font></i></a>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
;<br/><i><font color="Green">E183</font></i>: <font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="cat_2.html#K12">]</a></span> = 
<font color="Maroon">s</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">s</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">b</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="cat_2.html#T3">CAT_2:3</a></i>
<br/>.= 
<font color="Maroon">T34</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">T34</font> <a href="nattra_1.html#K8">`*`</a> <font color="Maroon">T13</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><font color="Maroon">T34</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">T13</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i>, <a class="txt" href="isocat_2.html#E7:31"><i><font color="Green">E55</font></i></a>, <a class="ref" href="nattra_1.html#T27">NATTRA_1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</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">fi</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">t</font> <a href="nattra_1.html#K5">.</a> <span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</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">fi</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">fi</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</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">c</font></span><a href="cat_2.html#K12">]</a></span></span>)</span>
<b>by </b><i>, , <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">fi</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</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">c</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">T24</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">T12</font> <a href="nattra_1.html#K5">.</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/>.= 
<span class="p1">(<span class="default"><font color="Maroon">T24</font> <a href="nattra_1.html#K8">`*`</a> <font color="Maroon">T12</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">c</font>
<b>by </b><i>, , <a class="ref" href="nattra_1.html#T27">NATTRA_1:27</a></i>
;<br/></div><b>end;</b></div><a NAME="E34:31_8"/><b>then </b><i><font color="Green">E184</font></i>: 
<font color="Maroon">T34</font> <a href="nattra_1.html#K8">`*`</a> <font color="Maroon">T13</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">T24</font> <a href="nattra_1.html#K8">`*`</a> <font color="Maroon">T12</font>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#T20">NATTRA_1:20</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</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">G</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</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="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="funct_2.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"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S4</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">T34</font> <a href="nattra_1.html#K8">`*`</a> <font color="Maroon">T13</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="isocat_2.html#E8:31"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="nattra_1.html#T42">NATTRA_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <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">G</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="nattra_1.html#T42">NATTRA_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</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">G</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</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"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</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">G</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E18:31"/><i><font color="Green">E185</font></i>: 
 <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_9"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_9"/>
 <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font> <a href="nattra_1.html#R1">is_transformable_to</a>  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a><br/>

<b>thus </b><a NAME="E2:31_9"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="nattra_1.html#M1">transformation</a> of  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font>, <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font> st <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>  holds <br/>(  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or for <font color="Olive">b<sub>4</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> holds  <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<a NAME="E19:31"/>
<font color="Maroon">G</font> is    <a href="nattra_1.html#M2">natural_transformation</a> of  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font>, <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_10"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_10"/>
 <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a>  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D8">NATTRA_1:def 8</a><br/>

<b>thus </b><a NAME="E2:31_10"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>  holds <br/>(  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> holds  <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as    <a href="nattra_1.html#M2">natural_transformation</a> of  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font>, <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font> ;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>let </b><font color="Maroon">s</font> be   <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/>

<b>assume </b><a NAME="E21:31"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">t</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">s</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/>

<b>thus </b><font color="Maroon">G</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="isocat_2.html#T7" target="_self">Th7</a>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="cat_2.html#K12">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span></span><a href="cat_2.html#K12">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K2">curry</a> <font color="Maroon">s</font></span>)</span> <a href="cat_2.html#K1">.</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, </i>
;<br/>


</div>
