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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">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">F3</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">F2</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">e</font> be    <a href="functor3.html#M1" target="_self">natural_equivalence</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font>;<br/><b>let </b><font color="Maroon">e1</font> be    <a href="functor3.html#M1" target="_self">natural_equivalence</a> of <font color="Maroon">F2</font>,<font color="Maroon">F3</font>;<br/><b>let </b><font color="Maroon">k</font> be    <a href="functor3.html#M1" target="_self">natural_equivalence</a> of <font color="Maroon">F1</font>,<font color="Maroon">F3</font>;<br/>















<b>assume </b><b>that </b><br/><a NAME="E1:62"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">k</font> <a href="pboole.html#R6">=</a> <font color="Maroon">e1</font> <a href="functor2.html#K5">`*`</a> <font color="Maroon">e</font>
 <b>and </b><br/><a NAME="E2:62"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">F1</font>,<font color="Maroon">F2</font> <a href="functor3.html#R1" target="_self">are_naturally_equivalent</a> 
 <b>and </b><br/><a NAME="E3:62"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">F2</font>,<font color="Maroon">F3</font> <a href="functor3.html#R1" target="_self">are_naturally_equivalent</a> 
 ;<br/>

<a NAME="E4:62"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F3</font>,<font color="Maroon">F1</font> <a href="functor3.html#R1" target="_self">are_naturally_equivalent</a> 
 
<b>by </b><i><a class="txt" href="functor3.html#E5:62"><i><font color="Green">E35</font></i></a>, , </i>;<br/>
<a NAME="E5:62"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">F3</font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">F2</font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F1</font> )
 
<b>by </b><i><a class="txt" href="functor3.html#E5:62"><i><font color="Green">E35</font></i></a>, , </i>;<br/>
<a NAME="E6:62"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">F3</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">F2</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">F1</font> )
 
<b>by </b><i><a class="txt" href="functor3.html#E5:62"><i><font color="Green">E35</font></i></a>, , </i>;<br/>
<a NAME="E7:62"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">F3</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">F1</font>
 
<b>by </b><i><a class="ref" href="functor2.html#T4">FUNCTOR2:4</a></i>;<br/>
<a NAME="E8:62"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">F1</font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">F2</font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F3</font> )
 
<b>by </b><i><a class="txt" href="functor3.html#E5:62"><i><font color="Green">E35</font></i></a>, , </i>;<br/>
<a NAME="E9:62"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">F1</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">F2</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">F3</font> )
 
<b>by </b><i><a class="txt" href="functor3.html#E5:62"><i><font color="Green">E35</font></i></a>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_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="E1:62_1"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">e</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font> is <a href="altcat_3.html#V3">iso</a> &amp; <font color="Maroon">e1</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#E5:62"><i><font color="Green">E35</font></i></a>, , </i>;<br/><a NAME="E2:62_1"/><i><font color="Green">E106</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F3</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F3</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="functor0.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, , <a class="ref" href="functor2.html#D1">FUNCTOR2:def 1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="functor3.html#K8" target="_self">"</a> </span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e1</font> <a href="functor2.html#K5">`*`</a> <font color="Maroon">e</font></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span> <a href="altcat_3.html#K1">"</a> 
<b>by </b><i><a class="txt" href="functor3.html#E4:62"><i><font color="Green">E34</font></i></a>, , <a class="txt" href="functor3.html#E2:62"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e1</font> <a href="functor2.html#K3">`*`</a> <font color="Maroon">e</font></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span> <a href="altcat_3.html#K1">"</a> 
<b>by </b><i>, <a class="ref" href="functor2.html#D8">FUNCTOR2:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e1</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="altcat_3.html#K1">"</a> 
<b>by </b><i>, <a class="ref" href="functor2.html#D5">FUNCTOR2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span> <a href="altcat_3.html#K1">"</a> </span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e1</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span> <a href="altcat_3.html#K1">"</a> </span>)</span>
<b>by </b><i>, <a class="txt" href="functor3.html#E6:62"><i><font color="Green">E44</font></i></a>, <a class="ref" href="altcat_3.html#T7">ALTCAT_3:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span> <a href="altcat_3.html#K1">"</a> </span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e1</font> <a href="functor3.html#K8" target="_self">"</a> </span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="txt" href="functor3.html#E2:62"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="functor3.html#K8" target="_self">"</a> </span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e1</font> <a href="functor3.html#K8" target="_self">"</a> </span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E5:62"><i><font color="Green">E35</font></i></a>, <a class="txt" href="functor3.html#E2:62"><i><font color="Green">E32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="functor3.html#K8" target="_self">"</a> </span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">e1</font> <a href="functor3.html#K8" target="_self">"</a> </span>)</span></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="functor2.html#D5">FUNCTOR2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="functor3.html#K8" target="_self">"</a> </span>)</span> <a href="functor2.html#K5">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">e1</font> <a href="functor3.html#K8" target="_self">"</a> </span>)</span></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="functor3.html#E4"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="functor2.html#D8">FUNCTOR2:def 8</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:62"/>
<font color="Maroon">k</font> <a href="functor3.html#K8" target="_self">"</a>  <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="functor3.html#K8" target="_self">"</a> </span>)</span> <a href="functor2.html#K5">`*`</a> <span class="p1">(<span class="default"><font color="Maroon">e1</font> <a href="functor3.html#K8" target="_self">"</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="functor2.html#T5">FUNCTOR2:5</a></i>;<br/>


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