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

<b>let </b><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">D</font>, <font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font><br/> for <font color="Olive" title="b3">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b4">H</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font><br/> for <font color="Olive" title="b5">s</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font>  st <font color="Olive" title="b1">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Olive" title="b3">G</font></span>)</span> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Olive" title="b5">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b4">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">G</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Olive" title="b5">s</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">F1</font>, <font color="Maroon" title="c6">F2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">H</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font><br/> for <font color="Olive" title="b3">s</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font>  st <font color="Maroon" title="c5">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">F2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Olive" title="b1">G</font></span>)</span> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Olive" title="b3">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Olive" title="b3">s</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c7">G</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">H</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font><br/> for <font color="Olive" title="b2">s</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font>  st <font color="Maroon" title="c5">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">F2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Olive" title="b2">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Olive" title="b2">s</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c8">H</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font>  st <font color="Maroon" title="c5">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">F2</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Olive" title="b1">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Olive" title="b1">s</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c9">s</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">F2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">s</font></span>)</span> )</span><br/>

















<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c5">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">F2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">s</font></span>)</span></span><br/>

<a NAME="E2:39"/><b>then </b><i><font color="Green" title="E31">A2</font></i>: 
( <font color="Maroon" title="c7">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">F2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">F2</font> )
 
<b>by </b><i><a class="ref" href="isocat_1.html#T27" target="_self" title="ISOCAT_1:th.27">Th27</a></i>;<br/>
<a NAME="E3:39"/><i><font color="Green" title="E32">A3</font></i>: 
( <font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F1</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F1</font> &amp; <font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">F2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">F2</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">v</font> = <font color="Maroon" title="c8">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">s</font></span>)</span> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">F2</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c3">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">s</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c11">a</font></span><br/><a NAME="E1:39_1"/><i><font color="Green" title="E33">A4</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="isocat_1.html#T30" target="_self" title="ISOCAT_1:th.30">Th30</a></i>;<br/><a NAME="E2:39_1"/><i><font color="Green" title="E34">A5</font></i>: 
( <font color="Maroon" title="c7">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">F1</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">a</font> &amp; <font color="Maroon" title="c7">G</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">F2</font></span>)</span> <a href="cat_1.html#K9" title="CAT_1:func.9">.</a> <font color="Maroon" title="c11">a</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T113" title="CAT_1:th.113">CAT_1:113</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">s</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c11">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="isocat_1.html#T28" target="_self" title="ISOCAT_1:th.28">Th28</a></i>
<br/>.= 
<font color="Maroon" title="c8">H</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">s</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:39_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="nattra_1.html#T11" title="NATTRA_1:th.11">NATTRA_1:11</a></i>
<br/>.= 
<font color="Maroon" title="c8">H</font> <a href="nattra_1.html#K3" title="NATTRA_1:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">s</font></span>)</span> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c11">a</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="#E2:39_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="isocat_1.html#T28" target="_self" title="ISOCAT_1:th.28">Th28</a></i>
<br/>.= 
<font color="Maroon" title="c10">v</font> <a href="nattra_1.html#K5" title="NATTRA_1:func.5">.</a> <font color="Maroon" title="c11">a</font>
<b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="#E3:39"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="isocat_1.html#T27" target="_self" title="ISOCAT_1:th.27">Th27</a>, <a class="ref" href="isocat_1.html#T28" target="_self" title="ISOCAT_1:th.28">Th28</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:39"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">H</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">H</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="isocat_1.html#K4" title="ISOCAT_1:func.4">*</a> <font color="Maroon" title="c9">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="isocat_1.html#T31" target="_self" title="ISOCAT_1:th.31">Th31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
