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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> ;<br/>
<div><i><font color="Green">E7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>consider </b><font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">s</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E2:59_1_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">t</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">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E3:59_1_1_1"/>
<font color="Maroon">F1</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E15:59_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a>  = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">F1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><b>hence </b><a NAME="E5:59_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">t</font>) <a href="hidden.html#R2">in</a>  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="cat_2.html#D2">CAT_2:def 2</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">d</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>, <a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font><b> such that </b><br/><a NAME="E3:59_1_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> holds  <font color="Maroon">d</font> <a href="cat_2.html#K6">.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8">FUNCT_2:sch 8</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> ;<br/>
<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>consider </b><font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">s</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E2:59_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">t</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">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E3:59_1_1_2"/>
<font color="Maroon">F1</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E15:59_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a>  = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">F2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><b>hence </b><a NAME="E5:59_1_1_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">t</font>) <a href="hidden.html#R2">in</a>  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="cat_2.html#D2">CAT_2:def 2</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>, <a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font><b> such that </b><br/><a NAME="E6:59_1_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> holds  <font color="Maroon">c</font> <a href="cat_2.html#K6">.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">t</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8">FUNCT_2:sch 8</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">F</font>, <font color="Olive">F1</font>, <font color="Olive">F2</font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> ex <font color="Olive">t</font> being   <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Olive">F</font>,<font color="Olive">F1</font> ex <font color="Olive">t1</font> being   <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Olive">F1</font>,<font color="Olive">F2</font> st <br/>( <font color="Maroon">a<sub>2</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="Olive">F</font>,<font color="Olive">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">t</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a<sub>1</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="Olive">F1</font>,<font color="Olive">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">t1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a<sub>3</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="Olive">F</font>,<font color="Olive">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive">t1</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Olive">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span> );<br/>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:59_1_1_3"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> )
 ;<br/><b>assume </b><a NAME="E2:59_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z</font>]
 ;<br/><b>then consider </b><font color="Maroon">F</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F</font>,<font color="Maroon">F1</font>, <font color="Maroon">t1</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E4:59_1_1_3"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">y</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">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</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">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">z</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">F</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><a NAME="E5:59_1_1_3"/>
( <font color="Maroon">F</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">F1</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F2</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E6:59_1_1_3"/><b>then </b>
<font color="Maroon">F</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:59_1_1_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:59_1_1_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> )
 ;<br/><b>assume </b><a NAME="E2:59_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z1</font>]
 ;<br/><b>then consider </b><font color="Maroon">F</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F</font>,<font color="Maroon">F1</font>, <font color="Maroon">t1</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E4:59_1_1_4"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">y</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">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</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">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">z1</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">F</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><b>assume </b><a NAME="E5:59_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">z2</font>]
 ;<br/><b>then consider </b><font color="Maroon">F'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F1'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F'</font>,<font color="Maroon">F1'</font>, <font color="Maroon">t1'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1'</font>,<font color="Maroon">F2'</font><b> such that </b><br/><a NAME="E7:59_1_1_4"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">y</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">F'</font>,<font color="Maroon">F1'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">x</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">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">z2</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">F'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t1'</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 ;<br/><a NAME="E8:59_1_1_4"/><i><font color="Green">E40</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F'</font>,<font color="Maroon">F1'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">t</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">t'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">t1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">t1'</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E9:59_1_1_4"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F'</font> &amp; <font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F1'</font> &amp; <font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F2'</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E10:59_1_1_4"/>
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">m</font> being   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font><b> such that </b><br/><a NAME="E10:59_1_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">m</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> &amp;  ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Olive">z</font>] ) )
 <b>and </b><br/><a NAME="E11:59_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>,<font color="Maroon">m</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span>]
 <b>from </b><i><a class="ref" href="partfun1.html#S4">PARTFUN1:sch 4</a>(, );<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="cat_2.html#M2">Element</a> of  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</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">a<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" target="_self">id</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E12:59_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">F</font> being   <a href="cat_2.html#M2">Element</a> of  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> holds  <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">F</font>) <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E15:59_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>, <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font><b> such that </b><br/><a NAME="E14:59_1_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">F</font> being   <a href="cat_2.html#M2">Element</a> of  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> holds  <font color="Maroon">i</font> <a href="funct_2.html#K3">.</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">F</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8">FUNCT_2:sch 8</a>();<br/></i>
<b>set </b><font color="Maroon">C</font> =  <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #);<br/>
<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_5"><b>now </b></a><div class="add"><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">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">t</font> be    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F</font>,<font color="Maroon">F1</font>;<br/><b>let </b><font color="Maroon">t1</font> be    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font>;<br/><b>assume </b><a NAME="E1:59_1_1_5"/>
( <font color="Maroon">F1</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">F</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F1</font> )
 ;<br/><a NAME="E2:59_1_1_5"/><b>then </b><i><font color="Green">E46</font></i>: 
( <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">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> &amp; <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">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E2:59_1_1_5"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E3:59_1_1_5"/>
<font color="Maroon">S<sub>1</sub></font>[<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">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span>,<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">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span>,<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">F</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span>]
 ;<br/><a NAME="E4:59_1_1_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">m</font>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">F'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F1'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F'</font>,<font color="Maroon">F1'</font>, <font color="Maroon">t1'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1'</font>,<font color="Maroon">F2'</font><b> such that </b><br/><a NAME="E6:59_1_1_5"/><i><font color="Green">E47</font></i>: 
( <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">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <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">F'</font>,<font color="Maroon">F1'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t'</font></span><a href="tarski.html#K4">]</a></span> &amp; <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">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span> <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">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1'</font></span><a href="tarski.html#K4">]</a></span> )
 <b>and </b><br/><a NAME="E7:59_1_1_5"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">m</font> <a href="funct_1.html#K1">.</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"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <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">F'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t1'</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E8:59_1_1_5"/><i><font color="Green">E50</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F'</font>,<font color="Maroon">F1'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">t</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">t'</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">t1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">t1'</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E9:59_1_1_5"/><b>then </b>
( <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F'</font> &amp; <font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F1'</font> &amp; <font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F2'</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E10:59_1_1_5"/>
<font color="Maroon">m</font> <a href="funct_1.html#K1">.</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"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <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">F</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_6"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:59_1_1_6"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) iff the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #), <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #);<br/>

<b>thus </b><a NAME="E1:59_1_1_6_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) implies the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_6_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:59_1_1_6_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #)
 ;<br/>

<b>then consider </b><font color="Maroon">F</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F</font>,<font color="Maroon">F1</font>, <font color="Maroon">t1</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E3:59_1_1_6_1_1"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">f</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">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">g</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">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span> )
 <b>and </b><br/><a NAME="E4:59_1_1_6_1_1"/>
<font color="Maroon">m</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <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">F</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>
<b>thus </b>the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">F1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>

<b>consider </b><font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E3:59_1_1_6_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">g</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">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">F1</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F2</font> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E15:59_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">F1'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1'</font>,<font color="Maroon">F2'</font><b> such that </b><br/><a NAME="E5:59_1_1_6_1"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">f</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">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">F1'</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F2'</font> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E15:59_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>assume </b><a NAME="E6:59_1_1_6_1"/><i><font color="Green">E95</font></i>: 
the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
 ;<br/>

<i><font color="Green">E96</font></i>: <font color="Maroon">F1</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">F2'</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F2'</font>,<font color="Maroon">F2</font> ;<br/>
<a NAME="E9:59_1_1_6_1"/>
<font color="Maroon">m</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <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">F1'</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E10:59_1_1_6_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #)
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:59_1_1_6"/>
for <font color="Olive">f</font>, <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #)  st the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/>( the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> &amp; the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #), <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #);<br/>

<b>assume </b><a NAME="E1:59_1_1_6_2"/>
the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:59_1_1_6_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">F</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F</font>,<font color="Maroon">F1</font>, <font color="Maroon">t1</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E4:59_1_1_6_2"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">f</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">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">g</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">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">m</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <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">F</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">F1'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1'</font>,<font color="Maroon">F2'</font><b> such that </b><br/><a NAME="E6:59_1_1_6_2"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">f</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">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">F1'</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F2'</font> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E15:59_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E7:59_1_1_6_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E8:59_1_1_6_2"/><b>then </b><i><font color="Green">E99</font></i>: 
( <font color="Maroon">F</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F1'</font> &amp; <font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F2'</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon">F1'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1'</font>,<font color="Maroon">F2'</font><b> such that </b><br/><a NAME="E10:59_1_1_6_2"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">g</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">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">F1'</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F2'</font> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E15:59_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E11:59_1_1_6_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E12:59_1_1_6_2"/><b>then </b>
( <font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F1'</font> &amp; <font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F2'</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E13:59_1_1_6_2"/><b>then </b>
<font color="Maroon">F</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 
<b>by </b><i>, <a class="ref" href="nattra_1.html#T3" target="_self">Th3</a>, , </i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <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">F</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t1</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> <b>by </b><i/>;<br/>
<b>thus </b>the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i>, </i>
;<br/>

<b>thus </b>the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">F2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font>
<b>by </b><i>, </i>
;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:59_1_1_6"/>
for <font color="Olive">f</font>, <font color="Olive">g</font>, <font color="Olive">h</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #)  st the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> &amp; the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> holds <br/>the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">h</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">h</font>,<font color="Olive">g</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_6_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #), <font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #), <font color="Maroon">h</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #);<br/>

<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font>, <font color="Maroon">g'</font> = <font color="Maroon">g</font>, <font color="Maroon">h'</font> = <font color="Maroon">h</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> ;<br/>
<b>assume </b><a NAME="E2:59_1_1_6_3"/><i><font color="Green">E101</font></i>: 
( the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> &amp; the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E3:59_1_1_6_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g'</font>,<font color="Maroon">f'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F1'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t1</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F1'</font>, <font color="Maroon">t2</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1'</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E5:59_1_1_6_3"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">f'</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">F1</font>,<font color="Maroon">F1'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">g'</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">F1'</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">m</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g'</font>,<font color="Maroon">f'</font></span><a href="tarski.html#K4">]</a></span> <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">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t2</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t1</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E6:59_1_1_6_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h'</font>,<font color="Maroon">g'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">m</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">F2'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F3</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F3'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t2'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F2'</font>,<font color="Maroon">F3</font>, <font color="Maroon">t3</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F3</font>,<font color="Maroon">F3'</font><b> such that </b><br/><a NAME="E8:59_1_1_6_3"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">g'</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">F2'</font>,<font color="Maroon">F3</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t2'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">h'</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">F3</font>,<font color="Maroon">F3'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t3</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">m</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h'</font>,<font color="Maroon">g'</font></span><a href="tarski.html#K4">]</a></span> <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">F2'</font>,<font color="Maroon">F3'</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t3</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t2'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E9:59_1_1_6_3"/><i><font color="Green">E104</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F2'</font>,<font color="Maroon">F3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1'</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, <a class="ref" href="nattra_1.html#T3" target="_self">Th3</a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E10:59_1_1_6_3"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F3</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E11:59_1_1_6_3"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">g'</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">F1'</font>,<font color="Maroon">F3</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t2</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E12:59_1_1_6_3"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F1'</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:59_1_1_6_3"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">F1'</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F3</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E14:59_1_1_6_3"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">F3</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F3'</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E15:59_1_1_6_3"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F3</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#T5" target="_self">Th5</a>, <a class="ref" href="nattra_1.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E16:59_1_1_6_3"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">F1'</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F3'</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#T6" target="_self">Th6</a>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">t2</font> = <font color="Maroon">t2</font> as    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1'</font>,<font color="Maroon">F3</font> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>thus </b>the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<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">F1</font>,<font color="Maroon">F3'</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t3</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <span class="p3">(<span class="default"><font color="Maroon">t2</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t1</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="nattra_1.html#T3" target="_self">Th3</a>, , , </i>
<br/>.= 
<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">F1</font>,<font color="Maroon">F3'</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t3</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t2</font></span>)</span> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t1</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="nattra_1.html#T5" target="_self">Th5</a>, <a class="ref" href="nattra_1.html#T6" target="_self">Th6</a>, , <a class="txt" href="nattra_1.html#E6:59_1_1"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<font color="Maroon">m</font> <a href="funct_1.html#K1">.</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"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1'</font>,<font color="Maroon">F3'</font></span><a href="tarski.html#K4">]</a></span>,<span class="p3">(<span class="default"><font color="Maroon">t3</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t2</font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">f'</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="nattra_1.html#T5" target="_self">Th5</a>, </i>
<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<font color="Maroon">g</font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="nattra_1.html#T3" target="_self">Th3</a>, <a class="ref" href="nattra_1.html#T4" target="_self">Th4</a>, <a class="ref" href="nattra_1.html#T6" target="_self">Th6</a>, </i>
;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #);<br/><b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">b</font> as    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> <b>by </b><i><a class="ref" href="cat_2.html#D2">CAT_2:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">s</font> = <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">F</font>,<font color="Maroon">F</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" target="_self">id</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K4">]</a></span> as    <a href="subset_1.html#M1">Element</a> of  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> <b>by </b><i><a class="txt" href="nattra_1.html#E15:59_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E6:59_1_1_6"/><i><font color="Green">E171</font></i>: 
the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">b</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">F</font>,<font color="Maroon">F</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" target="_self">id</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><b>hence </b>the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><br/><b>thus </b>the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/><b>thus </b><a NAME="E9:59_1_1_6"/>
for <font color="Olive">f</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #)  st the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> holds <br/>the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<font color="Olive">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Olive">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_6_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #);<br/>

<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> ;<br/>
<b>consider </b><font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E3:59_1_1_6_6"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">f'</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">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:59_1_1_6_6"/>
<font color="Maroon">F1</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E15:59_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>assume </b><a NAME="E5:59_1_1_6_6"/><i><font color="Green">E173</font></i>: 
the <a href="cat_1.html#U4">Cod</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>

<i><font color="Green">E174</font></i>: <font color="Maroon">F2</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">F</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#T3" target="_self">Th3</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F</font> ;<br/>
<a NAME="E8:59_1_1_6_6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>,<font color="Maroon">f'</font>,<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">F1</font>,<font color="Maroon">F</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nattra_1.html#K7" target="_self">id</a> <font color="Maroon">F</font></span>)</span> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span>]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:59_1_1_6_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s</font>,<font color="Maroon">f'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:59_1_1_6_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">f'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">m</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">F'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F1'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F'</font>,<font color="Maroon">F1'</font>, <font color="Maroon">t1'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1'</font>,<font color="Maroon">F2'</font><b> such that </b><br/><a NAME="E12:59_1_1_6_6"/><i><font color="Green">E175</font></i>: 
( <font color="Maroon">f'</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">F'</font>,<font color="Maroon">F1'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F</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">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1'</font></span><a href="tarski.html#K4">]</a></span> )
 <b>and </b><br/><a NAME="E13:59_1_1_6_6"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">m</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F</font></span>)</span>,<font color="Maroon">f'</font></span><a href="tarski.html#K4">]</a></span> <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">F'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t1'</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>
<a NAME="E14:59_1_1_6_6"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">F'</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F1'</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:59_1_1_6_6"/>
<font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F</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">F</font>,<font color="Maroon">F</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" target="_self">id</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E16:59_1_1_6_6"/><b>then </b><i><font color="Green">E178</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">t1'</font> <a href="funct_2.html#R2">=</a>  <a href="nattra_1.html#K7" target="_self">id</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E17:59_1_1_6_6"/><b>then </b>
( <font color="Maroon">F1'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F2'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E18:59_1_1_6_6"/>
the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#T4" target="_self">Th4</a>, <a class="ref" href="nattra_1.html#T5" target="_self">Th5</a>, <a class="ref" href="nattra_1.html#T6" target="_self">Th6</a>, </i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon">g</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U2">Morphisms</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #);<br/><b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> ;<br/><b>consider </b><font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E12:59_1_1_6"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">g'</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">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E13:59_1_1_6"/>
<font color="Maroon">F1</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E15:59_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><b>assume </b><a NAME="E14:59_1_1_6"/><i><font color="Green">E180</font></i>: 
the <a href="cat_1.html#U3">Dom</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><i><font color="Green">E181</font></i>: <font color="Maroon">F1</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">F</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#T3" target="_self">Th3</a></i>
;<br/><b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F</font>,<font color="Maroon">F2</font> ;<br/><a NAME="E17:59_1_1_6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">g'</font>,<font color="Maroon">s</font>,<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">F</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <span class="p3">(<span class="default"><a href="nattra_1.html#K7" target="_self">id</a> <font color="Maroon">F</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>]
 <b>by </b><i>, </i>;<br/><a NAME="E18:59_1_1_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g'</font>,<font color="Maroon">s</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><a NAME="E19:59_1_1_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g'</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">F'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F1'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F'</font>,<font color="Maroon">F1'</font>, <font color="Maroon">t1'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1'</font>,<font color="Maroon">F2'</font><b> such that </b><br/><a NAME="E21:59_1_1_6"/><i><font color="Green">E182</font></i>: 
( <font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F</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">F'</font>,<font color="Maroon">F1'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">g'</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">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t1'</font></span><a href="tarski.html#K4">]</a></span> )
 <b>and </b><br/><a NAME="E22:59_1_1_6"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">m</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g'</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <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">F'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t1'</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t'</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E23:59_1_1_6"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">F1'</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F2'</font>
 <b>by </b><i>, </i>;<br/><a NAME="E24:59_1_1_6"/>
<font color="Maroon">i</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">F</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">F</font>,<font color="Maroon">F</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" target="_self">id</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E25:59_1_1_6"/><b>then </b><i><font color="Green">E185</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F'</font>,<font color="Maroon">F1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">F</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">t'</font> <a href="funct_2.html#R2">=</a>  <a href="nattra_1.html#K7" target="_self">id</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E26:59_1_1_6"/><b>then </b>
( <font color="Maroon">F1'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F'</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E27:59_1_1_6"/>
the <a href="cat_1.html#U5">Comp</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="nattra_1.html#T4" target="_self">Th4</a>, <a class="ref" href="nattra_1.html#T5" target="_self">Th5</a>, <a class="ref" href="nattra_1.html#T6" target="_self">Th6</a>, </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">C</font> =  <a href="cat_1.html#G1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<font color="Maroon">d</font>,<font color="Maroon">c</font>,<font color="Maroon">m</font>,<font color="Maroon">i</font> #) as  <a href="cat_1.html#V1">strict</a> <a href="cat_1.html#NM4">Category</a> <b>by </b><i><a class="ref" href="cat_1.html#D8">CAT_1:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<b>thus </b><a NAME="E18:59_1_1"/>
( the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> &amp; the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="nattra_1.html#K11" target="_self">NatTrans</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> )
 ;<br/>

<b>thus </b><a NAME="E19:59_1_1"/><i><font color="Green">E186</font></i>: 
for <font color="Olive">f</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font> holds <br/> (  <a href="cat_1.html#K2">dom</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a>  &amp;  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">f</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a>  )
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E20:59_1_1"/>
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st  <a href="cat_1.html#K2">dom</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">f</font> holds <br/><span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>, <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>;<br/>

<b>consider </b><font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E2:59_1_1_7"/><i><font color="Green">E187</font></i>: 
( <font color="Maroon">f</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">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">F1</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F2</font> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E15:59_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">F1'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">F2'</font> being    <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>, <font color="Maroon">t'</font> being    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F1'</font>,<font color="Maroon">F2'</font><b> such that </b><br/><a NAME="E4:59_1_1_7"/><i><font color="Green">E188</font></i>: 
( <font color="Maroon">g</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">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">F1'</font> <a href="nattra_1.html#R2" target="_self">is_naturally_transformable_to</a> <font color="Maroon">F2'</font> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E15:59_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>assume </b><a NAME="E5:59_1_1_7"/><i><font color="Green">E189</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
 ;<br/>

<i><font color="Green">E190</font></i>: <font color="Maroon">F2</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t'</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i>, , <a class="ref" href="nattra_1.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">F1'</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">t'</font> = <font color="Maroon">t'</font> as    <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Maroon">F2</font>,<font color="Maroon">F2'</font> ;<br/>
<a NAME="E8:59_1_1_7"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">g</font>,<font color="Maroon">f</font>,<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">F1</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">t'</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Maroon">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span>]
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E9:59_1_1_7"/>
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E21:59_1_1"/>
for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">C</font>  st <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> holds <br/> ex <font color="Olive">F</font>, <font color="Olive">F1</font>, <font color="Olive">F2</font> being   <a href="cat_1.html#M2">Functor</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> ex <font color="Olive">t</font> being   <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Olive">F</font>,<font color="Olive">F1</font> ex <font color="Olive">t1</font> being   <a href="nattra_1.html#M2" target="_self">natural_transformation</a> of <font color="Olive">F1</font>,<font color="Olive">F2</font> st <br/>( <font color="Olive">f</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="Olive">F</font>,<font color="Olive">F1</font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">t</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Olive">g</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="Olive">F1</font>,<font color="Olive">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Olive">t1</font></span><a href="tarski.html#K4">]</a></span> &amp; the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Olive">g</font>,<font color="Olive">f</font></span><a href="cat_2.html#K13">]</a></span> <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="Olive">F</font>,<font color="Olive">F2</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive">t1</font> <a href="nattra_1.html#K8" target="_self">`*`</a> <font color="Olive">t</font></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">C</font>;<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>assume </b><a NAME="E22:59_1_1"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F</font> as    <a href="cat_2.html#M2">Element</a> of  <a href="cat_2.html#K7">Funct</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> <b>by </b><i><a class="ref" href="cat_2.html#D2">CAT_2:def 2</a></i>;<br/>
<b>thus </b><a NAME="E24:59_1_1"/>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">a</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">F</font>,<font color="Maroon">F</font></span><a href="tarski.html#K4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" target="_self">id</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, </i>;<br/>


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