<?xml version="1.0"?>
<div class="add">

<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> 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>6</sub></font> be  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="functor2.html#M2">natural_transformation</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>













<b>assume </b><a NAME="E1:45"/><i><font color="Green">E23</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>
 ;<br/>

<a NAME="E2:45"/>
<font color="Maroon">c<sub>6</sub></font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="functor2.html#T9">FUNCTOR2:9</a></i>;<br/>
<b>hence </b><a NAME="E3:45"/>
<font color="Maroon">c<sub>6</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>6</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#E1:45"><i><font color="Green">E23</font></i></a>, <a class="txt" href="functor3.html#E32"><i><font color="Green">Lemma21</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D7">FUNCTOR2:def 7</a><br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E4:45"/><i><font color="Green">E24</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></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> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>;<br/>

<a NAME="E5:45"/><i><font color="Green">E25</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>
 
<b>by </b><i><a class="txt" href="functor3.html#E1:45"><i><font color="Green">E23</font></i></a>, <a class="ref" href="functor2.html#D6">FUNCTOR2:def 6</a></i>;<br/>
<a NAME="E6:45"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/>
<a NAME="E7:45"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/>
<a NAME="E8:45"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/>
<a NAME="E9:45"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/>
<a NAME="E10:45"/><i><font color="Green">E30</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">c<sub>4</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>9</sub></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="txt" href="functor3.html#E4:45"><i><font color="Green">E24</font></i></a>, <a class="ref" href="functor0.html#D19">FUNCTOR0:def 19</a></i>;<br/>
<a NAME="E11:45"/><i><font color="Green">E31</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">c<sub>4</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></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="txt" href="functor3.html#E5:45"><i><font color="Green">E25</font></i></a>, <a class="ref" href="functor2.html#D1">FUNCTOR2:def 1</a></i>;<br/>
<a NAME="E12:45"/><i><font color="Green">E32</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">c<sub>4</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>9</sub></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="txt" href="functor3.html#E5:45"><i><font color="Green">E25</font></i></a>, <a class="ref" href="functor2.html#D1">FUNCTOR2:def 1</a></i>;<br/>
<a NAME="E13:45"/><i><font color="Green">E33</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">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>9</sub></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="txt" href="functor3.html#E4:45"><i><font color="Green">E24</font></i></a>, <a class="ref" href="functor0.html#D19">FUNCTOR0:def 19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <b>by </b><i><a class="txt" href="functor3.html#E6:45"><i><font color="Green">E26</font></i></a>, <a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> = 
<font color="Maroon">c<sub>11</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E5:45"><i><font color="Green">E25</font></i></a>, <a class="txt" href="functor3.html#E9:45"><i><font color="Green">E29</font></i></a>, <a class="ref" href="functor3.html#T11" target="_self">Th11</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="functor3.html#E4:45"><i><font color="Green">E24</font></i></a>, <a class="ref" href="functor3.html#T6" target="_self">Th6</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E6:45"><i><font color="Green">E26</font></i></a>, <a class="txt" href="functor3.html#E7:45"><i><font color="Green">E27</font></i></a>, <a class="txt" href="functor3.html#E10:45"><i><font color="Green">E30</font></i></a>, <a class="txt" href="functor3.html#E12:45"><i><font color="Green">E32</font></i></a>, <a class="ref" href="functor0.html#D24">FUNCTOR0:def 24</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E1:45"><i><font color="Green">E23</font></i></a>, <a class="txt" href="functor3.html#E4:45"><i><font color="Green">E24</font></i></a>, <a class="ref" href="functor2.html#D7">FUNCTOR2:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K6">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K6">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E11:45"><i><font color="Green">E31</font></i></a>, <a class="txt" href="functor3.html#E13:45"><i><font color="Green">E33</font></i></a>, <a class="ref" href="functor0.html#D24">FUNCTOR0:def 24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="functor3.html#E4:45"><i><font color="Green">E24</font></i></a>, <a class="txt" href="functor3.html#E8:45"><i><font color="Green">E28</font></i></a>, <a class="txt" href="functor3.html#E9:45"><i><font color="Green">E29</font></i></a>, <a class="ref" href="functor3.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>10</sub></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">c<sub>6</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E5:45"><i><font color="Green">E25</font></i></a>, <a class="txt" href="functor3.html#E6:45"><i><font color="Green">E26</font></i></a>, <a class="ref" href="functor3.html#T11" target="_self">Th11</a></i>
;<br/>


</div>
