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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="functor2.html#M2">natural_transformation</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="functor2.html#M2">natural_transformation</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="functor2.html#M1">transformation</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="functor2.html#M1">transformation</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>

























<b>assume </b><b>that </b><br/><a NAME="E1:47"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>and </b><br/><a NAME="E2:47"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E3:47"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E4:47"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="functor3.html#E2:47"><i><font color="Green">E25</font></i></a>, <a class="txt" href="functor3.html#E3:47"><i><font color="Green">E26</font></i></a>, <a class="ref" href="functor2.html#T10">FUNCTOR2:10</a></i>;<br/>
<a NAME="E5:47"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="functor3.html#E1:47"><i><font color="Green">E24</font></i></a>, <a class="txt" href="functor3.html#E2:47"><i><font color="Green">E25</font></i></a>, <a class="txt" href="functor3.html#E3:47"><i><font color="Green">E26</font></i></a>, <a class="ref" href="functor2.html#D6">FUNCTOR2:def 6</a></i>;<br/>
<a NAME="E6:47"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>6</sub></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">E30</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="functor3.html#E5:47"><i><font color="Green">E28</font></i></a>, <a class="ref" href="functor3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E8:47"/>
<font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="functor3.html#E5:47"><i><font color="Green">E28</font></i></a>, <a class="ref" href="functor2.html#T4">FUNCTOR2:4</a></i>;<br/>
<a NAME="E9:47"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="functor3.html#E5:47"><i><font color="Green">E28</font></i></a>, <a class="ref" href="functor3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E10:47"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="functor3.html#E5:47"><i><font color="Green">E28</font></i></a>, <a class="ref" href="functor3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E11:47"/>
<font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="functor3.html#E1:47"><i><font color="Green">E24</font></i></a>, <a class="txt" href="functor3.html#E2:47"><i><font color="Green">E25</font></i></a>, <a class="txt" href="functor3.html#E32"><i><font color="Green">Lemma21</font></i></a></i>;<br/>
<a NAME="E12:47"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></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">E34</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="functor3.html#K7" target="_self">(#)</a> <font color="Maroon">c<sub>13</sub></font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="functor3.html#K7" target="_self">(#)</a> <font color="Maroon">c<sub>12</sub></font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="functor3.html#E2:47"><i><font color="Green">E25</font></i></a>, <a class="txt" href="functor3.html#E3:47"><i><font color="Green">E26</font></i></a>, <a class="txt" href="functor3.html#E5:47"><i><font color="Green">E28</font></i></a>, <a class="ref" href="functor3.html#T22" target="_self">Th22</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="functor2.html#K5">`*`</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="functor3.html#K7" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="functor2.html#K3">`*`</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="functor2.html#K3">`*`</a> <font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font> <a href="functor2.html#K5">`*`</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E4:47"><i><font color="Green">E27</font></i></a>, <a class="txt" href="functor3.html#E6:47"><i><font color="Green">E29</font></i></a>, <a class="ref" href="functor3.html#T22" target="_self">Th22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font> <a href="functor2.html#K5">`*`</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E5:47"><i><font color="Green">E28</font></i></a>, <a class="ref" href="functor3.html#T13" target="_self">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font> <a href="functor2.html#K3">`*`</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E2:47"><i><font color="Green">E25</font></i></a>, <a class="txt" href="functor3.html#E3:47"><i><font color="Green">E26</font></i></a>, <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">c<sub>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E5:47"><i><font color="Green">E28</font></i></a>, <a class="ref" href="functor3.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E7:47"><i><font color="Green">E30</font></i></a>, <a class="txt" href="functor3.html#E9:47"><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">c<sub>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>13</sub></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">c<sub>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E7:47"><i><font color="Green">E30</font></i></a>, <a class="txt" href="functor3.html#E9:47"><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">c<sub>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="functor3.html#K7" target="_self">(#)</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E3:47"><i><font color="Green">E26</font></i></a>, <a class="txt" href="functor3.html#E5:47"><i><font color="Green">E28</font></i></a>, <a class="ref" href="functor3.html#T22" target="_self">Th22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>13</sub></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">c<sub>11</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E9:47"><i><font color="Green">E31</font></i></a>, <a class="txt" href="functor3.html#E10:47"><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">c<sub>11</sub></font> <a href="functor3.html#K7" target="_self">(#)</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="functor3.html#K7" target="_self">(#)</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E7:47"><i><font color="Green">E30</font></i></a>, <a class="txt" href="functor3.html#E10:47"><i><font color="Green">E32</font></i></a>, <a class="txt" href="functor3.html#E12:47"><i><font color="Green">E33</font></i></a>, <a class="txt" href="functor3.html#E13:47"><i><font color="Green">E34</font></i></a>, <a class="ref" href="functor2.html#T8">FUNCTOR2:8</a></i>
;<br/>


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