<?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> 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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</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="c1">A</font>,<font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b3">e</font> being   <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font>  st <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">e</font> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a> </span>)</span> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">e</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">F1</font>, <font color="Maroon" title="c4">F2</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="c1">A</font>,<font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">e</font> being   <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font>  st <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">e</font> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a> </span>)</span> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">e</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">e</font> be    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a> </span>)</span> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">e</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:61"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a> </span>)</span> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">e</font></span><br/>

<a NAME="E2:61"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c3">F1</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c4">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor3.html#D4" target="_self" title="FUNCTOR3:def.4">Def4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a> </span>)</span> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a> </span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c6">a</font></span><br/><a NAME="E1:61_1"/><span class="lab"><font color="Green" title="E36">A3</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="c3">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></span>;<br/><a NAME="E2:61_1"/>
<font color="Maroon" title="c4">F2</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c3">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="functor3.html#D4" target="_self" title="FUNCTOR3:def.4">Def4</a></span>;<br/><a NAME="E3:61_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A4</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="c4">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></span>;<br/><a NAME="E4:61_1"/>
<font color="Maroon" title="c5">e</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c6">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="#E1:61"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="functor3.html#D5" target="_self" title="FUNCTOR3:def.5">Def5</a></span>;<br/><a NAME="E5:61_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A5</font></span>: 
( <font color="Maroon" title="c5">e</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c6">a</font> is  <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a>  &amp; <font color="Maroon" title="c5">e</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c6">a</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_3.html#T5" title="ALTCAT_3:th.5">ALTCAT_3:5</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a> </span>)</span> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a> </span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c6">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a> </span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="functor3.html#T38" target="_self" title="FUNCTOR3:th.38">Th38</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="functor3.html#T38" target="_self" title="FUNCTOR3:th.38">Th38</a></span>
<br/>.= 
<font color="Maroon" title="c5">e</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c6">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E3:61_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E5:61_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="altcat_3.html#T3" title="ALTCAT_3:th.3">ALTCAT_3:3</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:61"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">e</font> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a> </span>)</span> <a href="functor3.html#K8" title="FUNCTOR3:func.8">"</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="functor2.html#T5" title="FUNCTOR2:th.5">FUNCTOR2:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
