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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" target="_self">Functors</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>







<b>assume </b><a NAME="E1:61"/><i><font color="Green">E49</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:61"/>
the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12" target="_self">Functors</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18" target="_self">Def18</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E4:61"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="nattra_1.html#D15" target="_self">Def15</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><font color="Maroon">c<sub>3</sub></font> = 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="nattra_1.html#E1:61"><i><font color="Green">E49</font></i></a>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="nattra_1.html#E4:61"><i><font color="Green">E50</font></i></a>, <a class="ref" href="nattra_1.html#T39" target="_self">Th39</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>4</sub></font> = 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="nattra_1.html#E1:61"><i><font color="Green">E49</font></i></a>, <a class="ref" href="cat_1.html#T23">CAT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="nattra_1.html#E4:61"><i><font color="Green">E50</font></i></a>, <a class="ref" href="nattra_1.html#T39" target="_self">Th39</a></i>
;<br/>

<b>thus </b><a NAME="E7:61"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E4:61"><i><font color="Green">E50</font></i></a></i>;<br/>


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