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<div class="add">

<b>let </b><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> being <a href="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">A</font>,<font color="Maroon" title="c3">B</font><br/> for <font color="Olive" title="b3">G1</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="c3">B</font>,<font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b4">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/>( <font color="Olive" title="b3">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b3">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Olive" title="b2">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Olive" title="b3">G1</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Olive" title="b4">e</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Olive" title="b3">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b3">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Olive" title="b2">F2</font> )</span><br/><b>let </b><font color="Maroon" title="c4">F1</font>, <font color="Maroon" title="c5">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="c2">A</font>,<font color="Maroon" title="c3">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G1</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="c3">B</font>,<font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">e</font> being   <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font>  st <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  holds <br/>( <font color="Olive" title="b1">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Olive" title="b1">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Olive" title="b1">G1</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Olive" title="b2">e</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Olive" title="b1">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Olive" title="b1">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> )</span><br/><b>let </b><font color="Maroon" title="c6">G1</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="c3">B</font>,<font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">e</font> being   <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font>  st <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  holds <br/>( <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Olive" title="b1">e</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> )</span><br/><b>let </b><font color="Maroon" title="c7">e</font> be    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  implies ( <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Maroon" title="c7">e</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> ) )</span><br/>













<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Maroon" title="c7">e</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> )</span><br/>

<a NAME="E2:53"/><b>then </b><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c4">F1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c5">F2</font>
 
<b>by </b><i><a class="ref" href="functor3.html#D4" target="_self" title="FUNCTOR3:def.4">Def4</a></i>;<br/>
<a NAME="E3:53"/><i><font color="Green" title="E32">A3</font></i>: 
<font color="Maroon" title="c4">F1</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c5">F2</font>
 
<b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="functor3.html#D4" target="_self" title="FUNCTOR3:def.4">Def4</a></i>;<br/>
<a NAME="E4:53"/><i><font color="Green" title="E33">A4</font></i>: 
<font color="Maroon" title="c5">F2</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c4">F1</font>
 
<b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="functor3.html#D4" target="_self" title="FUNCTOR3:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">k</font> = <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Maroon" title="c7">e</font> as    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <b>by </b><i><a class="txt" href="#E2:53"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="functor3.html#T28" target="_self" title="FUNCTOR3:th.28">Th28</a></i>;<br/>
<div><i><font color="Green" title="E34">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">k</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a></span><br/><a NAME="E1:53_1"/><i><font color="Green" title="E35">A6</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span> )
 <b>by </b><i><a class="ref" href="functor0.html#T34" title="FUNCTOR0:th.34">FUNCTOR0:34</a></i>;<br/><a NAME="E2:53_1"/><i><font color="Green" title="E36">A7</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G1</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">e</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:53"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="functor3.html#T11" target="_self" title="FUNCTOR3:th.11">Th11</a></i>;<br/><a NAME="E3:53_1"/><i><font color="Green" title="E37">A8</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">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>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="#E3:53"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="#E4:53"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></i>;<br/><a NAME="E4:53_1"/>
<font color="Maroon" title="c7">e</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="functor3.html#D5" target="_self" title="FUNCTOR3:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E5:53_1"/>
<font color="Maroon" title="c8">k</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 <b>by </b><i><a class="txt" href="#E1:53_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="#E2:53_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="#E3:53_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="ref" href="altcat_4.html#T20" title="ALTCAT_4:th.20">ALTCAT_4:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E7:53"/>
<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<a NAME="E1:53_2"/>
<font color="Maroon" title="c6">G1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">G1</font>
 
<b>by </b><i><a class="ref" href="functor2.html#T9" title="FUNCTOR2:th.9">FUNCTOR2:9</a></i>;<br/>
<b>hence </b><a NAME="E2:53_2"/>
<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font>
 <b>by </b><i><a class="txt" href="#E2:53"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="functor3.html#E32"><i><font color="Green" title="E21">Lm2</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor3.html#D4" target="_self" title="FUNCTOR3:def.4">FUNCTOR3:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font> &amp;  ex <font color="Olive" title="b1">t</font> being   <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> st <br/>for <font color="Olive" title="b2">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Olive" title="b1">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> )</span><br/>

<b>thus </b><a NAME="E3:53_2"/>
<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>
 <b>by </b><i><a class="txt" href="#E4:53"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="functor3.html#T10" target="_self" title="FUNCTOR3:th.10">Th10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being   <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> st <br/>for <font color="Olive" title="b2">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Olive" title="b1">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a></span><br/>

<b>take </b>
<font color="Maroon" title="c8">k</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Maroon" title="c8">k</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></span><br/>

<b>let </b><font color="Maroon" title="c9">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">k</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a></span><br/>

<b>thus </b><a NAME="E4:53_2"/>
<font color="Maroon" title="c8">k</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 <b>by </b><i><a class="txt" href="#E6:53"><i><font color="Green" title="E34">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:53"/>
( <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor3.html#R1" title="FUNCTOR3:pred.1">are_naturally_equivalent</a>  &amp; <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Maroon" title="c7">e</font> is    <a href="functor3.html#M1" title="FUNCTOR3:mode.1">natural_equivalence</a> of <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> )
 <b>by </b><i><a class="txt" href="#E6:53"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="functor3.html#D5" target="_self" title="FUNCTOR3:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
