<?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>, <font color="Maroon">c<sub>4</sub></font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><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> 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>9</sub></font>, <font color="Maroon">c<sub>10</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>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>11</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>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="functor2.html#M1">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>13</sub></font> be    <a href="functor2.html#M1">transformation</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>;<br/>























<b>assume </b><a NAME="E1:43"/><i><font color="Green">E21</font></i>: 
( <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>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>9</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/>

<a NAME="E2:43"/><b>then </b><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>10</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <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> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="ref" href="functor3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E3:43"/><b>then </b><i><font color="Green">E23</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="functor3.html#E1:43"><i><font color="Green">E21</font></i></a>, <a class="ref" href="functor3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E4:43"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</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>6</sub></font> &amp; <font color="Maroon">c<sub>7</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>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="functor3.html#E1:43"><i><font color="Green">E21</font></i></a>, <a class="ref" href="functor3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E5:43"/><i><font color="Green">E25</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="functor3.html#K6" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <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>12</sub></font></span>)</span> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="functor3.html#E1:43"><i><font color="Green">E21</font></i></a>, <a class="ref" href="functor3.html#T15" target="_self">Th15</a>, <a class="ref" href="functor3.html#T16" target="_self">Th16</a>, <a class="ref" href="functor3.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E6:43"/><i><font color="Green">E26</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="functor3.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="functor0.html#T33">FUNCTOR0:33</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="functor3.html#K7" target="_self">(#)</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="functor3.html#K7" target="_self">(#)</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>8</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="functor3.html#K7" target="_self">(#)</a> <font color="Maroon">c<sub>11</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <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></span>)</span> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>6</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#K1" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>6</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>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>6</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>9</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E2:43"><i><font color="Green">E22</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>13</sub></font> <a href="functor3.html#K6" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</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>9</sub></font> <a href="functor3.html#K5" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></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> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E3:43"><i><font color="Green">E23</font></i></a>, <a class="txt" href="functor3.html#E5:43"><i><font color="Green">E25</font></i></a>, <a class="txt" href="functor3.html#E6:43"><i><font color="Green">E26</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>13</sub></font> <a href="functor3.html#K6" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="functor2.html#K3">`*`</a> <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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E4:43"><i><font color="Green">E24</font></i></a>, <a class="ref" href="functor3.html#T13" target="_self">Th13</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="functor3.html#K7" target="_self">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="functor3.html#K7" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="functor3.html#K7" target="_self">(#)</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>

;<br/>


</div>
