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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">F</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">G3</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">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">t'</font> be    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">G2</font>,<font color="Maroon">G3</font>;<br/>

















<b>assume </b><a NAME="E1:37"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">G1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G2</font> &amp; <font color="Maroon">G2</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G3</font> )
 ;<br/>

<a NAME="E2:37"/><b>then </b><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">G3</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#T25">NATTRA_1:25</a></i>;<br/>
<a NAME="E3:37"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</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">F</font> &amp; <font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G3</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:37"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G3</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#T25">NATTRA_1:25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/><a NAME="E1:37_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">t'</font> <a href="nattra_1.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t'</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="txt" href="isocat_1.html#E2:37_1"><i><font color="Green">E29</font></i></a>, </i>;<br/><a NAME="E2:37_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">G1</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">G2</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G2</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">G3</font> <a href="cat_1.html#K13">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G3</font> <a href="cat_1.html#K14">*</a> <font color="Maroon">F</font></span>)</span> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T113">CAT_1:113</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t'</font> <a href="nattra_1.html#K8">`*`</a> <font color="Maroon">t</font></span>)</span> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon">t'</font> <a href="nattra_1.html#K8">`*`</a> <font color="Maroon">t</font></span>)</span> <a href="nattra_1.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="isocat_1.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">t'</font> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="isocat_1.html#E2:37_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="nattra_1.html#T27">NATTRA_1:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t'</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K8">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="isocat_1.html#T3" target="_self">Th3</a>, , <a class="ref" href="isocat_1.html#T4" target="_self">Th4</a>, <a class="ref" href="nattra_1.html#T27">NATTRA_1:27</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:37"/>
<span class="p1">(<span class="default"><font color="Maroon">t'</font> <a href="nattra_1.html#K8">`*`</a> <font color="Maroon">t</font></span>)</span> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">t'</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K8">`*`</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="isocat_1.html#K5" target="_self">*</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


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