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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="altcat_1.html#NM4">category</a>;<br/>

<b>assume </b><a NAME="E1:49"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="yellow18.html#R2" target="_self">are_opposite</a>  &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="yellow18.html#R2" target="_self">are_opposite</a>  )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:49"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E3:49"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>5</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="functor3.html#D4">FUNCTOR3:def 4</a><br/>

<b>given </b><font color="Maroon">c<sub>7</sub></font> being    <a href="functor2.html#M2">natural_transformation</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font><b> such that </b><a NAME="E4:49"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2">!</a> <font color="Olive">b<sub>1</sub></font> is <a href="altcat_3.html#V3">iso</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E5:49"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:49"><i><font color="Green">E25</font></i></a>, <a class="ref" href="functor2.html#D6">FUNCTOR2:def 6</a></i>;<br/>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:49_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/><b>hence </b><a NAME="E2:49_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/><b>hence </b><a NAME="E3:49_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green">E24</font></i></a>, <a class="ref" href="yellow18.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E4:49_1"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green">E24</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self">Th9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:49"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor2.html#R1">is_transformable_to</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D1">FUNCTOR2:def 1</a><br/>

<a NAME="E1:49_2"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E6:49"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:49_2"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</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="txt" href="yellow18.html#E5:49"><i><font color="Green">E28</font></i></a>, <a class="ref" href="functor2.html#D1">FUNCTOR2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green">E24</font></i></a>, <a class="ref" href="yellow18.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E9:49"/>
<font color="Maroon">c<sub>12</sub></font> is    <a href="functor2.html#M1">transformation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:49_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor2.html#R1">is_transformable_to</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow18.html#E7:49"><i><font color="Green">E30</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D2">FUNCTOR2:def 2</a><br/>

<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>set </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>;<br/>
<a NAME="E2:49_3"/>
( <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2">!</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yellow18.html#E5:49"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow18.html#E6:49"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow18.html#D5" target="_self">Def5</a>, <a class="ref" href="functor2.html#D4">FUNCTOR2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:49_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> is   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="functor2.html#M1">transformation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:49_4"/><i><font color="Green">E32</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>16</sub></font>;<br/><a NAME="E2:49_4"/><i><font color="Green">E33</font></i>: 
( <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green">E24</font></i></a>, <a class="ref" href="yellow18.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E3:49_4"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green">E24</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E4:49_4"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2">!</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2">!</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="yellow18.html#E5:49"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow18.html#E7:49"><i><font color="Green">E30</font></i></a>, <a class="txt" href="yellow18.html#E2:49_4"><i><font color="Green">E33</font></i></a>, <a class="ref" href="functor2.html#D4">FUNCTOR2:def 4</a></i>;<br/><a NAME="E5:49_4"/><i><font color="Green">E36</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span><a href="altcat_1.html#K1">^&gt;</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="txt" href="yellow18.html#E1:49_4"><i><font color="Green">E32</font></i></a>, <a class="txt" href="yellow18.html#E3:49_4"><i><font color="Green">E34</font></i></a>, <a class="ref" href="functor0.html#D19">FUNCTOR0:def 19</a></i>;<br/><a NAME="E6:49_4"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor0.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor0.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:49_4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="functor3.html#T7">FUNCTOR3:7</a></i>;<br/><a NAME="E7:49_4"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor0.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor0.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:49_4"><i><font color="Green">E32</font></i></a>, <a class="txt" href="yellow18.html#E3:49_4"><i><font color="Green">E34</font></i></a>, <a class="ref" href="functor3.html#T8">FUNCTOR3:8</a></i>;<br/><a NAME="E8:49_4"/><b>then </b><i><font color="Green">E37</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yellow18.html#E5:49_4"><i><font color="Green">E36</font></i></a>, <a class="ref" href="yellow18.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E9:49_4"/><i><font color="Green">E38</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span></span><a href="altcat_1.html#K1">^&gt;</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="txt" href="yellow18.html#E5:49"><i><font color="Green">E28</font></i></a>, <a class="ref" href="functor2.html#D1">FUNCTOR2:def 1</a></i>;<br/><a NAME="E10:49_4"/><i><font color="Green">E39</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow18.html#E6:49"><i><font color="Green">E29</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2">!</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yellow18.html#E4:49_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="yellow18.html#E5:49_4"><i><font color="Green">E36</font></i></a>, <a class="txt" href="yellow18.html#E8:49_4"><i><font color="Green">E37</font></i></a>, <a class="txt" href="yellow18.html#E9:49_4"><i><font color="Green">E38</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2">!</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K8">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E2:49"><i><font color="Green">E25</font></i></a>, <a class="txt" href="yellow18.html#E1:49_4"><i><font color="Green">E32</font></i></a>, <a class="txt" href="yellow18.html#E3:49_4"><i><font color="Green">E34</font></i></a>, <a class="ref" href="functor2.html#D7">FUNCTOR2:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yellow18.html#E4:49_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="yellow18.html#E5:49_4"><i><font color="Green">E36</font></i></a>, <a class="txt" href="yellow18.html#E8:49_4"><i><font color="Green">E37</font></i></a>, <a class="txt" href="yellow18.html#E9:49_4"><i><font color="Green">E38</font></i></a>, <a class="txt" href="yellow18.html#E10:49_4"><i><font color="Green">E39</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self">Th9</a></i>
;<br/><br/></div><b>end;</b></div>
<b>thus </b><a NAME="E12:49"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor2.html#R2">is_naturally_transformable_to</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="functor3.html#D4">FUNCTOR3:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:49_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor2.html#R1">is_transformable_to</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow18.html#E7:49"><i><font color="Green">E30</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D6">FUNCTOR2:def 6</a><br/>

<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>thus </b><a NAME="E2:49_5"/>
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="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <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="subset_1.html#M1">Element</a> of <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="functor2.html#K2">!</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K6">.</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="functor2.html#K2">!</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow18.html#E11:49"><i><font color="Green">E31</font></i></a></i>;<br/>


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

<a NAME="E13:49"/>
<font color="Maroon">c<sub>13</sub></font> is    <a href="functor2.html#M2">natural_transformation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:49_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor2.html#R2">is_naturally_transformable_to</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow18.html#E12:49"><i><font color="Green">E32</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D7">FUNCTOR2:def 7</a><br/>

<b>thus </b><a NAME="E2:49_6"/>
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="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <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="subset_1.html#M1">Element</a> of <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="functor2.html#K2">!</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K6">.</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="functor2.html#K2">!</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow18.html#E11:49"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="functor2.html#M2">natural_transformation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<b>thus </b><a NAME="E15:49"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor2.html#R1">is_transformable_to</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D1">FUNCTOR2:def 1</a><br/>

<a NAME="E1:49_7"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E6:49"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E2:49_7"/><b>then </b>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green">E24</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self">Th9</a></i>;<br/>
<b>hence </b><a NAME="E3:49_7"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</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="txt" href="yellow18.html#E3:49"><i><font color="Green">E26</font></i></a>, <a class="ref" href="functor2.html#D1">FUNCTOR2:def 1</a></i>;<br/>


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

<b>take </b>
<font color="Maroon">c<sub>14</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E16:49"/><i><font color="Green">E33</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor3.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor3.html#K2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E6:49"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E17:49"/>
<span class="p1">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green">E24</font></i></a>, <a class="ref" href="yellow18.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E18:49"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2">!</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E5:49"><i><font color="Green">E28</font></i></a>, <a class="txt" href="yellow18.html#E7:49"><i><font color="Green">E30</font></i></a>, <a class="ref" href="functor2.html#D4">FUNCTOR2:def 4</a></i>;<br/>
<a NAME="E19:49"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2">!</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> is <a href="altcat_3.html#V3">iso</a>
 
<b>by </b><i><a class="txt" href="yellow18.html#E4:49"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E20:49"/>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" target="_self">dualizing-func</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span></span><a href="altcat_1.html#K1">^&gt;</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="txt" href="yellow18.html#E3:49"><i><font color="Green">E26</font></i></a>, <a class="txt" href="yellow18.html#E5:49"><i><font color="Green">E28</font></i></a>, <a class="ref" href="functor2.html#D1">FUNCTOR2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E21:49"/>
<font color="Maroon">c<sub>14</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>15</sub></font> is <a href="altcat_3.html#V3">iso</a>
 <b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green">E24</font></i></a>, <a class="txt" href="yellow18.html#E16:49"><i><font color="Green">E33</font></i></a>, <a class="txt" href="yellow18.html#E18:49"><i><font color="Green">E34</font></i></a>, <a class="txt" href="yellow18.html#E19:49"><i><font color="Green">E35</font></i></a>, <a class="ref" href="yellow18.html#T24" target="_self">Th24</a></i>;<br/>


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