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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> 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>6</sub></font>, <font color="Maroon">c<sub>7</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>8</sub></font> be    <a href="functor3.html#M1" target="_self">natural_equivalence</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>9</sub></font> be    <a href="functor3.html#M1" target="_self">natural_equivalence</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>

















<b>assume </b><a NAME="E1:55"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#R1" target="_self">are_naturally_equivalent</a>  &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#R1" target="_self">are_naturally_equivalent</a>  )
 ;<br/>

<a NAME="E2:55"/><b>then </b><i><font color="Green">E32</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>6</sub></font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="functor3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:55"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>,<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="functor3.html#R1" target="_self">are_naturally_equivalent</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="functor3.html#R1" target="_self">are_naturally_equivalent</a>  )
 
<b>by </b><i><a class="txt" href="functor3.html#E1:55"><i><font color="Green">E31</font></i></a>, <a class="ref" href="functor3.html#T35" target="_self">Th35</a>, <a class="ref" href="functor3.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E4:55"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> &amp; <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="ref" href="functor3.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E5:55"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="functor3.html#M1" target="_self">natural_equivalence</a> of <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> is    <a href="functor3.html#M1" target="_self">natural_equivalence</a> of <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>,<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#E1:55"><i><font color="Green">E31</font></i></a>, <a class="ref" href="functor3.html#T35" target="_self">Th35</a>, <a class="ref" href="functor3.html#T36" target="_self">Th36</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> as    <a href="functor2.html#M2">natural_transformation</a> of <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</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#E2:55"><i><font color="Green">E32</font></i></a>, <a class="ref" href="functor3.html#T29" target="_self">Th29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font> as    <a href="functor2.html#M2">natural_transformation</a> of <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>,<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#E2:55"><i><font color="Green">E32</font></i></a>, <a class="ref" href="functor3.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E8:55"/>
<font color="Maroon">c<sub>10</sub></font> <a href="functor2.html#K5">`*`</a> <font color="Maroon">c<sub>11</sub></font> is    <a href="functor3.html#M1" target="_self">natural_equivalence</a> of <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</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#E3:55"><i><font color="Green">E33</font></i></a>, <a class="txt" href="functor3.html#E5:55"><i><font color="Green">E35</font></i></a>, <a class="ref" href="functor3.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E9:55"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</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="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> is    <a href="functor3.html#M1" target="_self">natural_equivalence</a> of <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</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#E4:55"><i><font color="Green">E34</font></i></a>, <a class="ref" href="functor2.html#D8">FUNCTOR2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E10:55"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>,<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="functor3.html#R1" target="_self">are_naturally_equivalent</a>  &amp; <font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K7" target="_self">(#)</a> <font color="Maroon">c<sub>8</sub></font> is    <a href="functor3.html#M1" target="_self">natural_equivalence</a> of <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>7</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#E3:55"><i><font color="Green">E33</font></i></a>, <a class="ref" href="functor3.html#T33" target="_self">Th33</a></i>;<br/>


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