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<b>let </b><font color="Maroon">A</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">D</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">F1</font> be    <a href="cat_1.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="cat_1.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="cat_1.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="cat_1.html#M2">Functor</a> of <font color="Maroon">B</font>,<font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">H1</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">C</font>,<font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">H2</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">C</font>,<font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">s</font> be    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font>;<br/><b>let </b><font color="Maroon">t</font> be    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">G1</font>,<font color="Maroon">G2</font>;<br/><b>let </b><font color="Maroon">u</font> be    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">H1</font>,<font color="Maroon">H2</font>;<br/>

























<b>assume </b><b>that </b><br/><a NAME="E1:49"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 <b>and </b><br/><a NAME="E2:49"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">G1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G2</font>
 <b>and </b><br/><a NAME="E3:49"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">H1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">H2</font>
 ;<br/>

<a NAME="E4:49"/><i><font color="Green">E23</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G1</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">H1</font> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G1</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">H1</font> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G2</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">H1</font> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">H2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G2</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">H2</font> <a href="cat_1.html#K14">*</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<a NAME="E5:49"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">u</font> <a href="isocat_1.html#K5" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">G2</font></span>)</span> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">H1</font> <a href="isocat_1.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F2</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="isocat_1.html#K4" target="_self">*</a> <font color="Maroon">t</font></span>)</span> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">H1</font> <a href="isocat_1.html#K4" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="isocat_1.html#K4" target="_self">*</a> <font color="Maroon">s</font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G1</font></span>)</span> <a href="isocat_1.html#K4" target="_self">*</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, , , <a class="txt" href="isocat_1.html#E6:49"><i><font color="Green">E29</font></i></a>, <a class="ref" href="isocat_1.html#T2" target="_self">Th2</a>, <a class="ref" href="isocat_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E6:49"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G1</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G1</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:49"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">H1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">H1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:49"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G1</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G2</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:49"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">H1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G2</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">H2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:49"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G2</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <span class="p1">(<span class="default"><font color="Maroon">H2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G2</font></span>)</span> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:49"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font>
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><font color="Maroon">u</font> <a href="isocat_1.html#K6" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K6" target="_self">(#)</a> <font color="Maroon">s</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="isocat_1.html#K5" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F2</font></span>)</span></span>)</span> <a href="nattra_1.html#K8">`*`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="isocat_1.html#K4" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F2</font></span>)</span></span>)</span> <a href="nattra_1.html#K8">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="isocat_1.html#K4" target="_self">*</a> <span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="isocat_1.html#K4" target="_self">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">G2</font></span>)</span> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F2</font></span>)</span> <a href="nattra_1.html#K8">`*`</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="isocat_1.html#K4" target="_self">*</a> <font color="Maroon">t</font></span>)</span> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F2</font></span>)</span></span>)</span> <a href="nattra_1.html#K8">`*`</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">G1</font></span>)</span> <a href="isocat_1.html#K4" target="_self">*</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="isocat_1.html#T8" target="_self">Th8</a>, , , , , <a class="ref" href="nattra_1.html#T28">NATTRA_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="isocat_1.html#K6" target="_self">(#)</a> <font color="Maroon">t</font></span>)</span> <a href="isocat_1.html#K6" target="_self">(#)</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="isocat_1.html#T9" target="_self">Th9</a>, , </i>
;<br/>


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