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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">F1</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">G1</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">C</font>, <font color="Maroon">G2</font> be    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">C</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:59"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G1</font>
 <b>and </b><br/><a NAME="E2:59"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">F2</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">G2</font>
 ;<br/>

<a NAME="E3:59"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">F1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">G1</font> &amp; <font color="Maroon">F2</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">G2</font> )
 
<b>by </b><i>, , <a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/>
<b>let </b><font color="Maroon">t1</font> be    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">G1</font>;<br/><b>let </b><font color="Maroon">t2</font> be    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F2</font>,<font color="Maroon">G2</font>;<br/>



<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">t1</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">A</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">B</font> ;<br/>
<a NAME="E5:59"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="isocat_2.html#K10" target="_self">&lt;:</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="isocat_2.html#K10" target="_self">:&gt;</a></span> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <span class="p1"><a href="isocat_2.html#K10" target="_self">&lt;:</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="isocat_2.html#K10" target="_self">:&gt;</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:59"/><b>then </b><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="isocat_2.html#K10" target="_self">&lt;:</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="isocat_2.html#K10" target="_self">:&gt;</a></span> <a href="nattra_1.html#R1">is_transformable_to</a> <span class="p1"><a href="isocat_2.html#K10" target="_self">&lt;:</a><span class="default"><font color="Maroon">G1</font>,<font color="Maroon">G2</font></span><a href="isocat_2.html#K10" target="_self">:&gt;</a></span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/>
<b>thus </b> <a href="isocat_2.html#K13" target="_self">Pr1</a> <span class="p1"><a href="isocat_2.html#K16" target="_self">&lt;:</a><span class="default"><font color="Maroon">t1</font>,<font color="Maroon">t2</font></span><a href="isocat_2.html#K16" target="_self">:&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="isocat_2.html#K8" target="_self">pr1</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> <a href="isocat_1.html#K2">*</a> <span class="p1"><a href="isocat_2.html#K16" target="_self">&lt;:</a><span class="default"><font color="Maroon">t1</font>,<font color="Maroon">t2</font></span><a href="isocat_2.html#K16" target="_self">:&gt;</a></span>
<b>by </b><i>, <a class="ref" href="isocat_1.html#D7">ISOCAT_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K8" target="_self">pr1</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1"><a href="isocat_2.html#K16" target="_self">&lt;:</a><span class="default"><font color="Maroon">t1</font>,<font color="Maroon">t2</font></span><a href="isocat_2.html#K16" target="_self">:&gt;</a></span>
<b>by </b><i>, <a class="ref" href="isocat_1.html#D5">ISOCAT_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K8" target="_self">pr1</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1"><a href="isocat_2.html#K15" target="_self">&lt;:</a><span class="default"><font color="Maroon">t1</font>,<font color="Maroon">t2</font></span><a href="isocat_2.html#K15" target="_self">:&gt;</a></span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K8" target="_self">pr1</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t2</font></span><a href="funct_3.html#K14">:&gt;</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K9">pr1</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">B</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t2</font></span><a href="funct_3.html#K14">:&gt;</a></span>
<b>by </b><i><a class="ref" href="cat_2.html#D10">CAT_2:def 10</a></i>
<br/>.= 
<font color="Maroon">t1</font>
<b>by </b><i><a class="ref" href="funct_3.html#T82">FUNCT_3:82</a></i>
;<br/>

<b>thus </b> <a href="isocat_2.html#K14" target="_self">Pr2</a> <span class="p1"><a href="isocat_2.html#K16" target="_self">&lt;:</a><span class="default"><font color="Maroon">t1</font>,<font color="Maroon">t2</font></span><a href="isocat_2.html#K16" target="_self">:&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="isocat_2.html#K9" target="_self">pr2</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> <a href="isocat_1.html#K2">*</a> <span class="p1"><a href="isocat_2.html#K16" target="_self">&lt;:</a><span class="default"><font color="Maroon">t1</font>,<font color="Maroon">t2</font></span><a href="isocat_2.html#K16" target="_self">:&gt;</a></span>
<b>by </b><i>, <a class="ref" href="isocat_1.html#D7">ISOCAT_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K9" target="_self">pr2</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1"><a href="isocat_2.html#K16" target="_self">&lt;:</a><span class="default"><font color="Maroon">t1</font>,<font color="Maroon">t2</font></span><a href="isocat_2.html#K16" target="_self">:&gt;</a></span>
<b>by </b><i>, <a class="ref" href="isocat_1.html#D5">ISOCAT_1:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K9" target="_self">pr2</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1"><a href="isocat_2.html#K15" target="_self">&lt;:</a><span class="default"><font color="Maroon">t1</font>,<font color="Maroon">t2</font></span><a href="isocat_2.html#K15" target="_self">:&gt;</a></span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="isocat_2.html#K9" target="_self">pr2</a> <font color="Maroon">B</font>,<font color="Maroon">C</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t2</font></span><a href="funct_3.html#K14">:&gt;</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K10">pr2</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">B</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1"><a href="funct_3.html#K14">&lt;:</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">t2</font></span><a href="funct_3.html#K14">:&gt;</a></span>
<b>by </b><i><a class="ref" href="cat_2.html#D11">CAT_2:def 11</a></i>
<br/>.= 
<font color="Maroon">t2</font>
<b>by </b><i><a class="ref" href="funct_3.html#T82">FUNCT_3:82</a></i>
;<br/>


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