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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">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">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">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">G3</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">s</font> be    <a href="functor2.html#M2">natural_transformation</a> of <font color="Maroon">G1</font>,<font color="Maroon">G2</font>;<br/><b>let </b><font color="Maroon">s1</font> be    <a href="functor2.html#M2">natural_transformation</a> of <font color="Maroon">G2</font>,<font color="Maroon">G3</font>;<br/><b>let </b><font color="Maroon">t</font> be    <a href="functor2.html#M1">transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font>;<br/><b>let </b><font color="Maroon">t1</font> be    <a href="functor2.html#M1">transformation</a> of <font color="Maroon">F2</font>,<font color="Maroon">F3</font>;<br/>

























<b>assume </b><b>that </b><br/><a NAME="E1:47"/><i><font color="Green">E31</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>and </b><br/><a NAME="E2:47"/><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>and </b><br/><a NAME="E3:47"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">G2</font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G3</font>
 ;<br/>

<a NAME="E4:47"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">G1</font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G3</font>
 
<b>by </b><i>, , <a class="ref" href="functor2.html#T10">FUNCTOR2:10</a></i>;<br/>
<a NAME="E5:47"/><i><font color="Green">E35</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">G1</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">G2</font> &amp; <font color="Maroon">F2</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">F3</font> &amp; <font color="Maroon">G2</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">G3</font> )
 
<b>by </b><i>, , , <a class="ref" href="functor2.html#D6">FUNCTOR2:def 6</a></i>;<br/>
<a NAME="E6:47"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">F1</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">F3</font>
 
<b>by </b><i><a class="ref" href="functor2.html#T4">FUNCTOR2:4</a></i>;<br/>
<a NAME="E7:47"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">G3</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F2</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">G3</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F3</font> &amp; <font color="Maroon">G3</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">G3</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:47"/>
<font color="Maroon">G1</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">G3</font>
 
<b>by </b><i>, <a class="ref" href="functor2.html#T4">FUNCTOR2:4</a></i>;<br/>
<a NAME="E9:47"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">G1</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">G3</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">G1</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">G2</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font> &amp; <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">G3</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F1</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:47"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">G2</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F2</font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">G3</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F2</font> &amp; <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">G2</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:47"/>
<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">F2</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E12:47"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">G1</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">G2</font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">F2</font>
 
<b>by </b><i><a class="ref" href="functor2.html#D6">FUNCTOR2:def 6</a></i>;<br/>
<a NAME="E13:47"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">s1</font> <a href="functor3.html#K7" target="_self">(#)</a> <font color="Maroon">t1</font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">G3</font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">t1</font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">F2</font></span>)</span> &amp; <font color="Maroon">s</font> <a href="functor3.html#K7" target="_self">(#)</a> <font color="Maroon">t</font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">t</font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">F1</font></span>)</span> )
 
<b>by </b><i>, , , </i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="functor2.html#K5">`*`</a> <font color="Maroon">s</font></span>)</span> <a href="functor3.html#K7" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">t1</font> <a href="functor2.html#K3">`*`</a> <font color="Maroon">t</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">G3</font> <a href="functor3.html#K5" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="functor2.html#K3">`*`</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="functor2.html#K5">`*`</a> <font color="Maroon">s</font></span>)</span> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">F1</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G3</font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">t1</font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">G3</font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="functor2.html#K5">`*`</a> <font color="Maroon">s</font></span>)</span> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">F1</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G3</font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">t1</font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">G3</font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="functor2.html#K3">`*`</a> <font color="Maroon">s</font></span>)</span> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">F1</font></span>)</span>
<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">G3</font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">t1</font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">G3</font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">F1</font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">F1</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G3</font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">t1</font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G3</font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">t</font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">F1</font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">F1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="functor3.html#E1:47"><i><font color="Green">E31</font></i></a>, <a class="ref" href="functor2.html#T8">FUNCTOR2:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G3</font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">t1</font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G3</font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">t</font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p3">(<span class="default"><font color="Maroon">s1</font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">F1</font></span>)</span></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">F1</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="functor3.html#E1:47"><i><font color="Green">E31</font></i></a>, <a class="ref" href="functor2.html#T8">FUNCTOR2:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G3</font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">t1</font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="functor3.html#K7" target="_self">(#)</a> <font color="Maroon">t</font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">F1</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G3</font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">t1</font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s1</font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">F2</font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G2</font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">t</font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">F1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E1:47"><i><font color="Green">E31</font></i></a>, <a class="txt" href="functor3.html#E2:47"><i><font color="Green">E32</font></i></a>, <a class="ref" href="functor2.html#T8">FUNCTOR2:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s1</font> <a href="functor3.html#K7" target="_self">(#)</a> <font color="Maroon">t1</font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="functor3.html#K7" target="_self">(#)</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i>, <a class="txt" href="functor3.html#E2:47"><i><font color="Green">E32</font></i></a>, <a class="txt" href="functor3.html#E3:47"><i><font color="Green">E33</font></i></a>, <a class="txt" href="functor3.html#E4:47"><i><font color="Green">E34</font></i></a>, <a class="ref" href="functor2.html#T8">FUNCTOR2:8</a></i>
;<br/>


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