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<b>let </b><font color="Maroon">A</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">F1</font> be  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">G1</font> be  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">G2</font> be  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="functor3.html#M1" target="_self">natural_equivalence</a> of <font color="Maroon">G1</font>,<font color="Maroon">G2</font>;<br/>













<b>assume </b><a NAME="E1:54"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">G1</font>,<font color="Maroon">G2</font> <a href="functor3.html#R1" target="_self">are_naturally_equivalent</a> 
 ;<br/>

<a NAME="E2:54"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">G1</font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G2</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">f</font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">F1</font> as    <a href="functor2.html#M2">natural_transformation</a> of <font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font>,<font color="Maroon">G2</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font> <b>by </b><i/>;<br/>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/><a NAME="E1:54_1"/><i><font color="Green">E34</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G1</font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G2</font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span> )
 <b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/><a NAME="E2:54_1"/>
<font color="Maroon">G1</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">G2</font>
 <b>by </b><i><a class="txt" href="functor3.html#E1:54"><i><font color="Green">E31</font></i></a>, </i>;<br/><a NAME="E3:54_1"/><b>then </b>
<font color="Maroon">k</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="functor2.html#K2">!</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:54_1"/>
<font color="Maroon">k</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font> is <a href="altcat_3.html#V3">iso</a>
 <b>by </b><i><a class="txt" href="functor3.html#E1:54"><i><font color="Green">E31</font></i></a>, <a class="txt" href="functor3.html#E1:54_1"><i><font color="Green">E34</font></i></a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E5:54"/>
<font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font>,<font color="Maroon">G2</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font> <a href="functor3.html#R1" target="_self">are_naturally_equivalent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>proof </b></a><div class="add">

<a NAME="E1:54_2"/>
<font color="Maroon">F1</font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F1</font>
 
<b>by </b><i><a class="ref" href="functor2.html#T9">FUNCTOR2:9</a></i>;<br/>
<b>hence </b><a NAME="E2:54_2"/>
<font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G2</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="txt" href="functor3.html#E2:54"><i><font color="Green">E32</font></i></a>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor3.html#D4" target="_self">FUNCTOR3:def 4</a><br/>

<a NAME="E3:54_2"/>
<font color="Maroon">G2</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">G1</font>
 
<b>by </b><i><a class="txt" href="functor3.html#E1:54"><i><font color="Green">E31</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:54_2"/>
<font color="Maroon">G2</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font>
 <b>by </b><i/>;<br/>

<b>take </b>
<font color="Maroon">k</font>
;<br/>

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

<b>thus </b><a NAME="E5:54_2"/>
<font color="Maroon">k</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font> is <a href="altcat_3.html#V3">iso</a>
 <b>by </b><i><a class="txt" href="functor3.html#E4:54"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:54"/>
( <font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font>,<font color="Maroon">G2</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font> <a href="functor3.html#R1" target="_self">are_naturally_equivalent</a>  &amp; <font color="Maroon">f</font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">F1</font> is    <a href="functor3.html#M1" target="_self">natural_equivalence</a> of <font color="Maroon">G1</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font>,<font color="Maroon">G2</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font> )
 <b>by </b><i><a class="txt" href="functor3.html#E4:54"><i><font color="Green">E33</font></i></a>, </i>;<br/>


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