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

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  implies  for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</font> being   <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>  st <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  holds <br/><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="Olive" title="b2">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="Olive" title="b1">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="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E2:48"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</font> being   <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>  st <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  holds <br/><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="Olive" title="b2">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="Olive" title="b1">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="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> </span><br/>

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c6">G</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  implies <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> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:48"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:48"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<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>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="functor3.html#D4" title="FUNCTOR3:def.4">FUNCTOR3:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  for <font color="Olive">b<sub>1</sub></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>  holds <br/> not  for <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="c2">B</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive">b<sub>2</sub></font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  or <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> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  )</span><br/>

<span class="kw">given </span><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><span class="kw"> such that </span><a NAME="E6:48"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
 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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> </span><br/>

<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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="E7:48"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="functor2.html#D6" title="FUNCTOR2:def.6">FUNCTOR2:def 6</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E29">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &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="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> &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="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> )</span><br/><a NAME="E1:48_1"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T33" title="FUNCTOR0:th.33">FUNCTOR0:33</a></span>;<br/><a NAME="E2:48_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="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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T33" title="FUNCTOR0:th.33">FUNCTOR0:33</a></span>;<br/><span class="kw">hence </span><a NAME="E3:48_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="ref" href="functor0.html#T33" title="FUNCTOR0:th.33">FUNCTOR0:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &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="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> )</span><br/><span class="kw">hence </span><a NAME="E4:48_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="yellow18.html#D5" target="_self" title="YELLOW18:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><span class="kw">hence </span><a NAME="E5:48_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="yellow18.html#T9" target="_self" title="YELLOW18:th.9">Th9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E9:48"/><span class="lab"><font color="Green" title="E30">A9</font></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="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="48_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <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>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E1:48_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48"><span class="lab"><font color="Green" title="E29">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:48_2"/>
 not <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>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">t</font> = 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>
<br/>.= 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="yellow18.html#D3" target="_self" title="YELLOW18:def.3">Def3</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">dt</font> = <font color="Maroon" title="c7">t</font> as   <a href="pboole.html#NM1" title="PBOOLE:NM.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> <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a>, <a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/>
<a NAME="E12:48"/>
<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="48_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:48_4"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E30">A9</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="functor2.html#D2" title="FUNCTOR2:def.2">FUNCTOR2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</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  <font color="Maroon" title="c12">dt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> is    <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"><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="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="yellow18.html#K2" 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="Olive">b<sub>1</sub></font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><br/>

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="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"><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></span><br/>

<span class="kw">set </span><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="E2:48_4"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="yellow18.html#D5" target="_self" title="YELLOW18:def.5">Def5</a></span>;<br/>
<a NAME="E3:48_4"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="functor2.html#D4" title="FUNCTOR2:def.4">FUNCTOR2:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:48_4"/>
<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="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"><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E2:48_4"><span class="lab"><font color="Green" title="E31">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><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><span class="lab"><font color="Green" title="E31">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>  implies  for <font color="Olive" title="b1">f</font> being   <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> 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="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="Olive" title="b1">f</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" title="b1">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> )</span><br/><span class="kw">assume </span><a NAME="E1:48_5"/><span class="lab"><font color="Green" title="E32">A12</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <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> 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="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="Olive" title="b1">f</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" title="b1">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></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> <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="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></span><br/><span class="kw">set </span><font color="Maroon" title="c17">b9</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/><span class="kw">set </span><font color="Maroon" title="c18">a9</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/><span class="kw">set </span><font color="Maroon" title="c19">f9</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="E2:48_5"/><span class="lab"><font color="Green" title="E33">A13</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="yellow18.html#D5" target="_self" title="YELLOW18:def.5">Def5</a></span>;<br/><a NAME="E3:48_5"/><span class="lab"><font color="Green" title="E34">A14</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="yellow18.html#D5" target="_self" title="YELLOW18:def.5">Def5</a></span>;<br/><a NAME="E4:48_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A15</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:48_5"><span class="lab"><font color="Green" title="E33">A13</font></span></a>, <a class="ref" href="yellow18.html#T9" target="_self" title="YELLOW18:th.9">Th9</a></span>;<br/><a NAME="E5:48_5"/><span class="lab"><font color="Green" title="E36">A16</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E2:48_5"><span class="lab"><font color="Green" title="E33">A13</font></span></a>, <a class="ref" href="functor2.html#D4" title="FUNCTOR2:def.4">FUNCTOR2:def 4</a></span>;<br/><a NAME="E6:48_5"/><span class="lab"><font color="Green" title="E37">A17</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E3:48_5"><span class="lab"><font color="Green" title="E34">A14</font></span></a>, <a class="ref" href="functor2.html#D4" title="FUNCTOR2:def.4">FUNCTOR2:def 4</a></span>;<br/><a NAME="E7:48_5"/><span class="lab"><font color="Green" title="E38">A18</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="functor2.html#D4" title="FUNCTOR2:def.4">FUNCTOR2:def 4</a></span>;<br/><a NAME="E8:48_5"/><span class="lab"><font color="Green" title="E39">A19</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="functor2.html#D4" title="FUNCTOR2:def.4">FUNCTOR2:def 4</a></span>;<br/><a NAME="E9:48_5"/><span class="lab"><font color="Green" title="E40">A20</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_5"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="txt" href="#E4:48_5"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="ref" href="functor0.html#D18" title="FUNCTOR0:def.18">FUNCTOR0:def 18</a></span>;<br/><a NAME="E10:48_5"/><span class="lab"><font color="Green" title="E41">A21</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_5"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="txt" href="#E4:48_5"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="ref" href="functor0.html#D18" title="FUNCTOR0:def.18">FUNCTOR0:def 18</a></span>;<br/><a NAME="E11:48_5"/><span class="lab"><font color="Green" title="E42">A22</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_5"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="ref" href="functor3.html#T7" title="FUNCTOR3:th.7">FUNCTOR3:7</a></span>;<br/><a NAME="E12:48_5"/><span class="lab"><font color="Green" title="E43">A23</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_5"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="ref" href="functor3.html#T7" title="FUNCTOR3:th.7">FUNCTOR3:7</a></span>;<br/><a NAME="E13:48_5"/><span class="lab"><font color="Green" title="E44">A24</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_5"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="txt" href="#E4:48_5"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="txt" href="#E11:48_5"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="ref" href="functor3.html#T8" title="FUNCTOR3:th.8">FUNCTOR3:8</a></span>;<br/><a NAME="E14:48_5"/><span class="lab"><font color="Green" title="E45">A25</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_5"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="txt" href="#E4:48_5"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="txt" href="#E12:48_5"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="ref" href="functor3.html#T8" title="FUNCTOR3:th.8">FUNCTOR3:8</a></span>;<br/><a NAME="E15:48_5"/><span class="lab"><font color="Green" title="E46">A26</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E9:48_5"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="txt" href="#E13:48_5"><span class="lab"><font color="Green" title="E44">A24</font></span></a>, <a class="ref" href="yellow18.html#D5" target="_self" title="YELLOW18:def.5">Def5</a></span>;<br/><a NAME="E16:48_5"/><span class="lab"><font color="Green" title="E47">A27</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E10:48_5"><span class="lab"><font color="Green" title="E41">A21</font></span></a>, <a class="txt" href="#E14:48_5"><span class="lab"><font color="Green" title="E45">A25</font></span></a>, <a class="ref" href="yellow18.html#D5" target="_self" title="YELLOW18:def.5">Def5</a></span>;<br/><a NAME="E17:48_5"/><span class="lab"><font color="Green" title="E48">A28</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></span>;<br/><a NAME="E18:48_5"/><span class="lab"><font color="Green" title="E49">A29</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></span>;<br/><a NAME="E19:48_5"/><span class="lab"><font color="Green" title="E50">A30</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48"><span class="lab"><font color="Green" title="E29">A7</font></span></a></span>;<br/><a NAME="E20:48_5"/><span class="lab"><font color="Green" title="E51">A31</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48"><span class="lab"><font color="Green" title="E29">A7</font></span></a></span>;<br/><a NAME="E21:48_5"/><span class="lab"><font color="Green" title="E52">A32</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48"><span class="lab"><font color="Green" title="E29">A7</font></span></a></span>;<br/><a NAME="E22:48_5"/><span class="lab"><font color="Green" title="E53">A33</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48"><span class="lab"><font color="Green" title="E29">A7</font></span></a></span>;<br/><span class="kw">hence </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E6:48_5"><span class="lab"><font color="Green" title="E37">A17</font></span></a>, <a class="txt" href="#E8:48_5"><span class="lab"><font color="Green" title="E39">A19</font></span></a>, <a class="txt" href="#E10:48_5"><span class="lab"><font color="Green" title="E41">A21</font></span></a>, <a class="txt" href="#E16:48_5"><span class="lab"><font color="Green" title="E47">A27</font></span></a>, <a class="txt" href="#E17:48_5"><span class="lab"><font color="Green" title="E48">A28</font></span></a>, <a class="txt" href="#E19:48_5"><span class="lab"><font color="Green" title="E50">A30</font></span></a>, <a class="txt" href="#E21:48_5"><span class="lab"><font color="Green" title="E52">A32</font></span></a>, <a class="ref" href="yellow18.html#T9" target="_self" title="YELLOW18:th.9">Th9</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E1:48_5"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="txt" href="#E4:48_5"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="ref" href="functor2.html#D7" title="FUNCTOR2:def.7">FUNCTOR2:def 7</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E5:48_5"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="txt" href="#E7:48_5"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="txt" href="#E9:48_5"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="txt" href="#E15:48_5"><span class="lab"><font color="Green" title="E46">A26</font></span></a>, <a class="txt" href="#E18:48_5"><span class="lab"><font color="Green" title="E49">A29</font></span></a>, <a class="txt" href="#E19:48_5"><span class="lab"><font color="Green" title="E50">A30</font></span></a>, <a class="txt" href="#E20:48_5"><span class="lab"><font color="Green" title="E51">A31</font></span></a>, <a class="txt" href="#E22:48_5"><span class="lab"><font color="Green" title="E53">A33</font></span></a>, <a class="ref" href="yellow18.html#T9" target="_self" title="YELLOW18:th.9">Th9</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E15:48"/>
<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>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="functor3.html#D4" title="FUNCTOR3:def.4">FUNCTOR3:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> &amp;  ex <font color="Olive">b<sub>1</sub></font> being    <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> st <br/> for <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  <font color="Olive">b<sub>1</sub></font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive">b<sub>2</sub></font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:48_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#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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E30">A9</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="functor2.html#D6" title="FUNCTOR2:def.6">FUNCTOR2:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being    <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> st <br/> for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being    <a href="subset_1.html#M1" 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>2</sub></font>,<font color="Olive">b<sub>3</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>4</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>2</sub></font>,<font color="Olive">b<sub>3</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="Olive">b<sub>1</sub></font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</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"><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>4</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>4</sub></font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c13">dt</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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> )</span><br/>

<span class="kw">thus </span><a NAME="E2:48_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:48"><span class="lab"><font color="Green" title="E31">A11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:48"><span class="lab"><font color="Green" title="E31">A11</font></span></a>, <a class="ref" href="functor2.html#D7" title="FUNCTOR2:def.7">FUNCTOR2:def 7</a></span>;<br/>
<span class="kw">thus </span><a NAME="E17:48"/>
<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>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being    <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> st <br/> for <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  <font color="Olive">b<sub>1</sub></font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive">b<sub>2</sub></font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <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>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E1:48_7"/><span class="lab"><font color="Green" title="E32">A34</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48"><span class="lab"><font color="Green" title="E29">A7</font></span></a></span>;<br/>
<a NAME="E2:48_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="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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48"><span class="lab"><font color="Green" title="E29">A7</font></span></a></span>;<br/>
<a NAME="E3:48_7"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E1:48_7"><span class="lab"><font color="Green" title="E32">A34</font></span></a>, <a class="ref" href="yellow18.html#T9" target="_self" title="YELLOW18:th.9">Th9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:48_7"/>
 not <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>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">take </span>
<font color="Maroon" title="c14">dt</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</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  <font color="Maroon" title="c14">dt</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive">b<sub>1</sub></font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> </span><br/>

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> </span><br/>

<a NAME="E18:48"/><span class="lab"><font color="Green" title="E32">A35</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48"><span class="lab"><font color="Green" title="E29">A7</font></span></a></span>;<br/>
<a NAME="E19:48"/><span class="lab"><font color="Green" title="E33">A36</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48"><span class="lab"><font color="Green" title="E29">A7</font></span></a></span>;<br/>
<a NAME="E20:48"/><span class="lab"><font color="Green" title="E34">A37</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="yellow18.html#D5" target="_self" title="YELLOW18:def.5">Def5</a></span>;<br/>
<a NAME="E21:48"/><span class="lab"><font color="Green" title="E35">A38</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="functor2.html#D4" title="FUNCTOR2:def.4">FUNCTOR2:def 4</a></span>;<br/>
<a NAME="E22:48"/><span class="lab"><font color="Green" title="E36">A39</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="txt" href="#E20:48"><span class="lab"><font color="Green" title="E34">A37</font></span></a>, <a class="ref" href="functor2.html#D4" title="FUNCTOR2:def.4">FUNCTOR2:def 4</a></span>;<br/>
<a NAME="E23:48"/><span class="lab"><font color="Green" title="E37">A40</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E27">A5</font></span></a></span>;<br/>
<a NAME="E24:48"/><span class="lab"><font color="Green" title="E38">A41</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></span>;<br/>
<a NAME="E25:48"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:48"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E18:48"><span class="lab"><font color="Green" title="E32">A35</font></span></a>, <a class="txt" href="#E19:48"><span class="lab"><font color="Green" title="E33">A36</font></span></a>, <a class="txt" href="#E21:48"><span class="lab"><font color="Green" title="E35">A38</font></span></a>, <a class="txt" href="#E22:48"><span class="lab"><font color="Green" title="E36">A39</font></span></a>, <a class="txt" href="#E23:48"><span class="lab"><font color="Green" title="E37">A40</font></span></a>, <a class="txt" href="#E24:48"><span class="lab"><font color="Green" title="E38">A41</font></span></a>, <a class="ref" href="yellow18.html#T23" target="_self" title="YELLOW18:th.23">Th24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
