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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font>, <font color="Maroon" title="c4">D</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>;<br/>

<b>assume </b><a NAME="E1:49"/><i><font color="Green" title="E24">A1</font></i>: 
( <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="yellow18.html#R2" title="YELLOW18:pred.2">are_opposite</a>  &amp; <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font> <a href="yellow18.html#R2" title="YELLOW18:pred.2">are_opposite</a>  )
 ;<br/>

<b>let </b><font color="Maroon" title="c5">F</font>, <font color="Maroon" title="c6">G</font> be  <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:49"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">G</font>
 <b>and </b><br/><a NAME="E3:49"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c6">G</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c5">F</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="functor3.html#D4" title="FUNCTOR3:def.4">FUNCTOR3:def 4</a><br/>

<b>given </b><font color="Maroon" title="c7">t</font> being    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c6">G</font><b> such that </b><a NAME="E5:49"/><i><font color="Green" title="E27">A4</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font> holds  <font color="Maroon" title="c7">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b1">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c8">dAB</font> =  <a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>;<br/>
<b>set </b><font color="Maroon" title="c9">dCD</font> =  <a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font>;<br/>
<b>set </b><font color="Maroon" title="c10">dF</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c11">dG</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>;<br/>
<a NAME="E6:49"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c6">G</font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:49"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="functor2.html#D6" title="FUNCTOR2:def.6">FUNCTOR2:def 6</a></i>;<br/>
<div><i><font color="Green" title="E29">A6</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" title="c12">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span> )
 <b>by </b><i><a class="ref" href="functor0.html#T34" title="FUNCTOR0:th.34">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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="functor0.html#T34" title="FUNCTOR0:th.34">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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="yellow18.html#D5" target="_self" title="YELLOW18:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E4:49_1"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self" title="YELLOW18:th.9">Th9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:49"/><i><font color="Green" title="E30">A7</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</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" title="c12">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a><br/>

<a NAME="E1:49_2"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E7:49"><i><font color="Green" title="E29">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:49_2"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="yellow18.html#E6:49"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c12">dt</font> = <font color="Maroon" title="c7">t</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="yellow18.html#D3" target="_self" title="YELLOW18:def.3">Def3</a></i>;<br/>
<a NAME="E10:49"/>
<font color="Maroon" title="c12">dt</font> is    <a href="functor2.html#M1" title="FUNCTOR2:mode.1">transformation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow18.html#E8:49"><i><font color="Green" title="E30">A7</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D2" title="FUNCTOR2:def.2">FUNCTOR2:def 2</a><br/>

<b>let </b><font color="Maroon" title="c13">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>;<br/>

<b>set </b><font color="Maroon" title="c14">b</font> = <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c13">a</font>;<br/>
<a NAME="E2:49_3"/>
( <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a</font> &amp; <font color="Maroon" title="c7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c13">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c13">a</font></span>)</span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c13">a</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c13">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c13">a</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c13">a</font></span>)</span></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="yellow18.html#E6:49"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="yellow18.html#E7:49"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="yellow18.html#D5" target="_self" title="YELLOW18:def.5">Def5</a>, <a class="ref" href="functor2.html#D4" title="FUNCTOR2:def.4">FUNCTOR2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:49_3"/>
<font color="Maroon" title="c12">dt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a</font> is   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c13">a</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c13">a</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c13">dt</font> = <font color="Maroon" title="c12">dt</font> as    <a href="functor2.html#M1" title="FUNCTOR2:mode.1">transformation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> ;<br/>
<div><i><font color="Green" title="E31">A8</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" title="c14">a</font>, <font color="Maroon" title="c15">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>;<br/><b>assume </b><a NAME="E1:49_4"/><i><font color="Green" title="E32">A9</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>let </b><font color="Maroon" title="c16">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font>;<br/><b>set </b><font color="Maroon" title="c17">b'</font> = <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">b</font>;<br/><b>set </b><font color="Maroon" title="c18">a'</font> = <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c14">a</font>;<br/><b>set </b><font color="Maroon" title="c19">f'</font> = <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c16">f</font>;<br/><a NAME="E2:49_4"/><i><font color="Green" title="E33">A10</font></i>: 
( <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> &amp; <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">b</font> )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="yellow18.html#D5" target="_self" title="YELLOW18:def.5">Def5</a></i>;<br/><a NAME="E3:49_4"/><b>then </b><i><font color="Green" title="E34">A11</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">b</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c14">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self" title="YELLOW18:th.9">Th9</a></i>;<br/><a NAME="E4:49_4"/><i><font color="Green" title="E35">A12</font></i>: 
( <font color="Maroon" title="c7">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c14">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> &amp; <font color="Maroon" title="c7">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b</font> &amp; <font color="Maroon" title="c13">dt</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> &amp; <font color="Maroon" title="c13">dt</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">b</font> )
 <b>by </b><i><a class="txt" href="yellow18.html#E6:49"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="yellow18.html#E8:49"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="yellow18.html#E2:49_4"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="functor2.html#D4" title="FUNCTOR2:def.4">FUNCTOR2:def 4</a></i>;<br/><a NAME="E5:49_4"/><i><font color="Green" title="E36">A13</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">b</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">b</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:49_4"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="yellow18.html#E3:49_4"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="functor0.html#D19" title="FUNCTOR0:def.19">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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c16">f</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c16">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:49_4"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="functor3.html#T7" title="FUNCTOR3:th.7">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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c16">f</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c16">f</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:49_4"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="yellow18.html#E3:49_4"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="functor3.html#T8" title="FUNCTOR3:th.8">FUNCTOR3:8</a></i>;<br/><a NAME="E8:49_4"/><b>then </b><i><font color="Green" title="E37">A14</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c16">f</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c16">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="yellow18.html#E5:49_4"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="yellow18.html#D5" target="_self" title="YELLOW18:def.5">Def5</a></i>;<br/><a NAME="E9:49_4"/><i><font color="Green" title="E38">A15</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">b</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">b</font></span>)</span></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="yellow18.html#E6:49"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></i>;<br/><a NAME="E10:49_4"/><i><font color="Green" title="E39">A16</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c14">a</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c14">a</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">b</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">b</font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow18.html#E7:49"><i><font color="Green" title="E29">A6</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c13">dt</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c15">b</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c16">f</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c16">f</font></span>)</span></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="yellow18.html#E4:49_4"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="yellow18.html#E5:49_4"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="yellow18.html#E8:49_4"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="yellow18.html#E9:49_4"><i><font color="Green" title="E38">A15</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self" title="YELLOW18:th.9">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c14">a</font></span>)</span></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K8" title="FUNCTOR0:func.8">.</a> <font color="Maroon" title="c16">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E2:49"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="yellow18.html#E1:49_4"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="yellow18.html#E3:49_4"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="functor2.html#D7" title="FUNCTOR2:def.7">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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c16">f</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">dt</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c14">a</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="yellow18.html#E4:49_4"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="yellow18.html#E5:49_4"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="yellow18.html#E8:49_4"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="yellow18.html#E9:49_4"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="yellow18.html#E10:49_4"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self" title="YELLOW18:th.9">Th9</a></i>
;<br/><br/></div><b>end;</b></div>
<b>thus </b><a NAME="E13:49"/><i><font color="Green" title="E32">A17</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="functor3.html#D4" title="FUNCTOR3:def.4">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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow18.html#E8:49"><i><font color="Green" title="E30">A7</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D6" title="FUNCTOR2:def.6">FUNCTOR2:def 6</a><br/>

<b>take </b>
<font color="Maroon" title="c13">dt</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" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>  holds <br/>( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&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" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&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" title="ALTCAT_1:func.1">^&gt;</a></span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">dt</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">dt</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow18.html#E12:49"><i><font color="Green" title="E31">A8</font></i></a></i>;<br/>


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

<a NAME="E14:49"/>
<font color="Maroon" title="c13">dt</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow18.html#E13:49"><i><font color="Green" title="E32">A17</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D7" title="FUNCTOR2:def.7">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" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>  holds <br/>( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&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" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&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" title="ALTCAT_1:func.1">^&gt;</a></span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c13">dt</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p3">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">dt</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow18.html#E12:49"><i><font color="Green" title="E31">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">dt</font> = <font color="Maroon" title="c13">dt</font> as    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> ;<br/>
<b>thus </b><a NAME="E16:49"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</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" title="c15">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E7:49"><i><font color="Green" title="E29">A6</font></i></a></i>;<br/>
<a NAME="E2:49_7"/><b>then </b>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</font></span>)</span></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self" title="YELLOW18:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E3:49_7"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="yellow18.html#E3:49"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></i>;<br/>


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

<b>take </b>
<font color="Maroon" title="c14">dt</font>
;<br/>

<b>let </b><font color="Maroon" title="c15">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>;<br/>

<a NAME="E17:49"/><i><font color="Green" title="E33">A18</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c6">G</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</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" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c4">D</font></span>)</span> <a href="functor3.html#K3" title="FUNCTOR3:func.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="functor3.html#K2" title="FUNCTOR3:func.2">*</a> <span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E7:49"><i><font color="Green" title="E29">A6</font></i></a></i>;<br/>
<a NAME="E18:49"/>
<span class="p1">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a</font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="yellow18.html#D5" target="_self" title="YELLOW18:def.5">Def5</a></i>;<br/>
<a NAME="E19:49"/><b>then </b><i><font color="Green" title="E34">A19</font></i>: 
( <font color="Maroon" title="c14">dt</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> &amp; <font color="Maroon" title="c7">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">a</font> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E6:49"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="yellow18.html#E8:49"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="functor2.html#D4" title="FUNCTOR2:def.4">FUNCTOR2:def 4</a></i>;<br/>
<a NAME="E20:49"/><i><font color="Green" title="E35">A20</font></i>: 
<font color="Maroon" title="c7">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</font></span>)</span> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 
<b>by </b><i><a class="txt" href="yellow18.html#E5:49"><i><font color="Green" title="E27">A4</font></i></a></i>;<br/>
<a NAME="E21:49"/>
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</font></span>)</span></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow18.html#K2" title="YELLOW18:func.2">dualizing-func</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c15">a</font></span>)</span></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="yellow18.html#E3:49"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="yellow18.html#E6:49"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E22:49"/>
<font color="Maroon" title="c14">dt</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c15">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 <b>by </b><i><a class="txt" href="yellow18.html#E1:49"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="yellow18.html#E17:49"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="yellow18.html#E19:49"><i><font color="Green" title="E34">A19</font></i></a>, <a class="txt" href="yellow18.html#E20:49"><i><font color="Green" title="E35">A20</font></i></a>, <a class="ref" href="yellow18.html#T24" target="_self" title="YELLOW18:th.24">Th24</a></i>;<br/>


</div>
