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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ;<br/>
<div><i><font color="Green" title="E47">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">t</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>;<br/><b>consider </b><font color="Maroon" title="c4">F1</font>, <font color="Maroon" title="c5">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c6">s</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font><b> such that </b><br/><a NAME="E2:58_1_1_1"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c6">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E3:58_1_1_1"/>
<font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c5">F2</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E2:58_1_1_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c4">F1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/><b>hence </b><a NAME="E5:58_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">t</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="cat_2.html#D2" title="CAT_2:def.2">CAT_2:def 2</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">d</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E3:58_1_1"/><i><font color="Green" title="E48">A3</font></i>: 
for <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> holds  <font color="Maroon" title="c3">d</font> <a href="cat_2.html#K6" title="CAT_2:func.6">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">t</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8" title="FUNCT_2:sch.8">FUNCT_2:sch 8</a>(<a class="txt" href="nattra_1.html#E1:58_1_1"><i><font color="Green" title="E47">A1</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> ;<br/>
<div><i><font color="Green" title="E49">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">t</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>;<br/><b>consider </b><font color="Maroon" title="c5">F1</font>, <font color="Maroon" title="c6">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c7">s</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font><b> such that </b><br/><a NAME="E2:58_1_1_2"/><i><font color="Green" title="E50">A5</font></i>: 
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c7">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E3:58_1_1_2"/>
<font color="Maroon" title="c5">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c6">F2</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">F1</font>,<font color="Maroon" title="c6">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E2:58_1_1_2"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c6">F2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/><b>hence </b><a NAME="E5:58_1_1_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">t</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="cat_2.html#D2" title="CAT_2:def.2">CAT_2:def 2</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">c</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E6:58_1_1"/><i><font color="Green" title="E50">A6</font></i>: 
for <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> holds  <font color="Maroon" title="c4">c</font> <a href="cat_2.html#K6" title="CAT_2:func.6">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">t</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8" title="FUNCT_2:sch.8">FUNCT_2:sch 8</a>(<a class="txt" href="nattra_1.html#E4:58_1_1"><i><font color="Green" title="E49">A4</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b4">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font> ex <font color="Olive" title="b5">t1</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b5">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b4">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> );<br/>
<div><i><font color="Green" title="E51">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:58_1_1_3"/><i><font color="Green" title="E52">A8</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> )
 ;<br/><b>assume </b><a NAME="E2:58_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font>]
 ;<br/><b>then consider </b><font color="Maroon" title="c8">F</font>, <font color="Maroon" title="c9">F1</font>, <font color="Maroon" title="c10">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c11">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font>, <font color="Maroon" title="c12">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font><b> such that </b><br/><a NAME="E4:58_1_1_3"/><i><font color="Green" title="E53">A9</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c11">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/><a NAME="E5:58_1_1_3"/>
( <font color="Maroon" title="c8">F</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c9">F1</font> &amp; <font color="Maroon" title="c9">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c10">F2</font> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E1:58_1_1_3"><i><font color="Green" title="E52">A8</font></i></a>, <a class="txt" href="nattra_1.html#E4:58_1_1_3"><i><font color="Green" title="E53">A9</font></i></a>, <a class="ref" href="nattra_1.html#T35" target="_self" title="NATTRA_1:th.35">Th35</a></i>;<br/><a NAME="E6:58_1_1_3"/><b>then </b>
<font color="Maroon" title="c8">F</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c10">F2</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#T25" target="_self" title="NATTRA_1:th.25">Th25</a></i>;<br/><b>hence </b><a NAME="E7:58_1_1_3"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E4:58_1_1_3"><i><font color="Green" title="E53">A9</font></i></a>, <a class="ref" href="nattra_1.html#T35" target="_self" title="NATTRA_1:th.35">Th35</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E52">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">z1</font>, <font color="Maroon" title="c8">z2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:58_1_1_4"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> )
 ;<br/><b>assume </b><a NAME="E2:58_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z1</font>]
 ;<br/><b>then consider </b><font color="Maroon" title="c9">F</font>, <font color="Maroon" title="c10">F1</font>, <font color="Maroon" title="c11">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c12">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c10">F1</font>, <font color="Maroon" title="c13">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font><b> such that </b><br/><a NAME="E4:58_1_1_4"/><i><font color="Green" title="E53">A11</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c10">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c13">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c7">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c11">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c12">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/><b>assume </b><a NAME="E5:58_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c8">z2</font>]
 ;<br/><b>then consider </b><font color="Maroon" title="c14">F'</font>, <font color="Maroon" title="c15">F1'</font>, <font color="Maroon" title="c16">F2'</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c17">t'</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c14">F'</font>,<font color="Maroon" title="c15">F1'</font>, <font color="Maroon" title="c18">t1'</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c15">F1'</font>,<font color="Maroon" title="c16">F2'</font><b> such that </b><br/><a NAME="E7:58_1_1_4"/><i><font color="Green" title="E54">A12</font></i>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F'</font>,<font color="Maroon" title="c15">F1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">t'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F1'</font>,<font color="Maroon" title="c16">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c18">t1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c8">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F'</font>,<font color="Maroon" title="c16">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">t1'</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c17">t'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/><a NAME="E8:58_1_1_4"/><i><font color="Green" title="E55">A13</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c10">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F'</font>,<font color="Maroon" title="c15">F1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c17">t'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F1'</font>,<font color="Maroon" title="c16">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">t1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c18">t1'</font> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E4:58_1_1_4"><i><font color="Green" title="E53">A11</font></i></a>, <a class="txt" href="nattra_1.html#E7:58_1_1_4"><i><font color="Green" title="E54">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E9:58_1_1_4"/><b>then </b>
( <font color="Maroon" title="c9">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c14">F'</font> &amp; <font color="Maroon" title="c10">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c15">F1'</font> &amp; <font color="Maroon" title="c11">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c16">F2'</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E10:58_1_1_4"/>
<font color="Maroon" title="c7">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z2</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E4:58_1_1_4"><i><font color="Green" title="E53">A11</font></i></a>, <a class="txt" href="nattra_1.html#E7:58_1_1_4"><i><font color="Green" title="E54">A12</font></i></a>, <a class="txt" href="nattra_1.html#E8:58_1_1_4"><i><font color="Green" title="E55">A13</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">m</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E10:58_1_1"/><i><font color="Green" title="E53">A14</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">m</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp;  ex <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] ) )
 <b>and </b><br/><a NAME="E11:58_1_1"/><i><font color="Green" title="E54">A15</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">m</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c5">m</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S5" title="BINOP_1:sch.5">BINOP_1:sch 5</a>(<a class="txt" href="nattra_1.html#E7:58_1_1"><i><font color="Green" title="E51">A7</font></i></a>, <a class="txt" href="nattra_1.html#E8:58_1_1"><i><font color="Green" title="E52">A10</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="cat_2.html#M2" title="CAT_2:mode.2">Element</a> of  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$1,$1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> $1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E12:58_1_1"/><i><font color="Green" title="E55">A16</font></i>: 
for <font color="Olive" title="b1">F</font> being   <a href="cat_2.html#M2" title="CAT_2:mode.2">Element</a> of  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> holds  <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">F</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">i</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E14:58_1_1"/><i><font color="Green" title="E56">A17</font></i>: 
for <font color="Olive" title="b1">F</font> being   <a href="cat_2.html#M2" title="CAT_2:mode.2">Element</a> of  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> holds  <font color="Maroon" title="c6">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">F</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8" title="FUNCT_2:sch.8">FUNCT_2:sch 8</a>(<a class="txt" href="nattra_1.html#E12:58_1_1"><i><font color="Green" title="E55">A16</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c7">C</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #);<br/>
<div><i><font color="Green" title="E57">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">F</font>, <font color="Maroon" title="c9">F1</font>, <font color="Maroon" title="c10">F2</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>;<br/><b>let </b><font color="Maroon" title="c11">t</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font>;<br/><b>let </b><font color="Maroon" title="c12">t1</font> be    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font>;<br/><b>assume </b><a NAME="E1:58_1_1_5"/>
( <font color="Maroon" title="c9">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c10">F2</font> &amp; <font color="Maroon" title="c8">F</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c9">F1</font> )
 ;<br/><a NAME="E2:58_1_1_5"/><b>then </b><i><font color="Green" title="E58">A19</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> )
 <b>by </b><i><a class="ref" href="nattra_1.html#T35" target="_self" title="NATTRA_1:th.35">Th35</a></i>;<br/><a NAME="E3:58_1_1_5"/>
<font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c11">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]
 ;<br/><a NAME="E4:58_1_1_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E10:58_1_1"><i><font color="Green" title="E53">A14</font></i></a>, <a class="txt" href="nattra_1.html#E2:58_1_1_5"><i><font color="Green" title="E58">A19</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">F'</font>, <font color="Maroon" title="c14">F1'</font>, <font color="Maroon" title="c15">F2'</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c16">t'</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c13">F'</font>,<font color="Maroon" title="c14">F1'</font>, <font color="Maroon" title="c17">t1'</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c14">F1'</font>,<font color="Maroon" title="c15">F2'</font><b> such that </b><br/><a NAME="E6:58_1_1_5"/><i><font color="Green" title="E59">A20</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F'</font>,<font color="Maroon" title="c14">F1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c16">t'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1'</font>,<font color="Maroon" title="c15">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">t1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>and </b><br/><a NAME="E7:58_1_1_5"/><i><font color="Green" title="E60">A21</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F'</font>,<font color="Maroon" title="c15">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">t1'</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c16">t'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E11:58_1_1"><i><font color="Green" title="E54">A15</font></i></a></i>;<br/><a NAME="E8:58_1_1_5"/><i><font color="Green" title="E61">A22</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F'</font>,<font color="Maroon" title="c14">F1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">t</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c16">t'</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1'</font>,<font color="Maroon" title="c15">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">t1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c17">t1'</font> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E6:58_1_1_5"><i><font color="Green" title="E59">A20</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E9:58_1_1_5"/><b>then </b>
( <font color="Maroon" title="c8">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c13">F'</font> &amp; <font color="Maroon" title="c9">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c14">F1'</font> &amp; <font color="Maroon" title="c10">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c15">F2'</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E10:58_1_1_5"/>
<font color="Maroon" title="c5">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F1</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c9">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">F</font>,<font color="Maroon" title="c10">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c11">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E7:58_1_1_5"><i><font color="Green" title="E60">A21</font></i></a>, <a class="txt" href="nattra_1.html#E8:58_1_1_5"><i><font color="Green" title="E61">A22</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:58_1_1_6"/><i><font color="Green" title="E58">A23</font></i>: 
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) iff the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #);<br/>

<b>thus </b><a NAME="E1:58_1_1_6_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) implies the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:58_1_1_6_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #)
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">F</font>, <font color="Maroon" title="c11">F1</font>, <font color="Maroon" title="c12">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c13">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c11">F1</font>, <font color="Maroon" title="c14">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c11">F1</font>,<font color="Maroon" title="c12">F2</font><b> such that </b><br/><a NAME="E3:58_1_1_6_1_1"/><i><font color="Green" title="E59">A24</font></i>: 
( <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c11">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c13">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">F1</font>,<font color="Maroon" title="c12">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c14">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>and </b><br/><a NAME="E4:58_1_1_6_1_1"/>
<font color="Maroon" title="c5">m</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c12">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c13">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E11:58_1_1"><i><font color="Green" title="E54">A15</font></i></a></i>;<br/>
<b>thus </b>the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_1"><i><font color="Green" title="E48">A3</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">F1</font>,<font color="Maroon" title="c12">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_1_6_1_1"><i><font color="Green" title="E59">A24</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c11">F1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c11">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_1_6_1_1"><i><font color="Green" title="E59">A24</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E6:58_1_1"><i><font color="Green" title="E50">A6</font></i></a></i>
;<br/>


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

<b>consider </b><font color="Maroon" title="c10">F1</font>, <font color="Maroon" title="c11">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c12">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font><b> such that </b><br/><a NAME="E3:58_1_1_6_1"/><i><font color="Green" title="E59">A25</font></i>: 
( <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">F2</font> )
 <b>by </b><i><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">F1'</font>, <font color="Maroon" title="c14">F2'</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c15">t'</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c13">F1'</font>,<font color="Maroon" title="c14">F2'</font><b> such that </b><br/><a NAME="E5:58_1_1_6_1"/><i><font color="Green" title="E60">A26</font></i>: 
( <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1'</font>,<font color="Maroon" title="c14">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">t'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">F1'</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c14">F2'</font> )
 <b>by </b><i><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></i>;<br/>
<b>assume </b><a NAME="E6:58_1_1_6_1"/><i><font color="Green" title="E61">A27</font></i>: 
the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font>
 ;<br/>

<i><font color="Green" title="E62">A28</font></i>: <font color="Maroon" title="c10">F1</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F1</font>,<font color="Maroon" title="c11">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_1_6_1"><i><font color="Green" title="E59">A25</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c4">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">f</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_1"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="nattra_1.html#E6:58_1_1_6_1"><i><font color="Green" title="E61">A27</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E6:58_1_1"><i><font color="Green" title="E50">A6</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1'</font>,<font color="Maroon" title="c14">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E5:58_1_1_6_1"><i><font color="Green" title="E60">A26</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c14">F2'</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">t</font> = <font color="Maroon" title="c12">t</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c14">F2'</font>,<font color="Maroon" title="c11">F2</font> ;<br/>
<a NAME="E9:58_1_1_6_1"/>
<font color="Maroon" title="c5">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1'</font>,<font color="Maroon" title="c11">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c15">t'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E15:58_1_1"><i><font color="Green" title="E57">A18</font></i></a>, <a class="txt" href="nattra_1.html#E3:58_1_1_6_1"><i><font color="Green" title="E59">A25</font></i></a>, <a class="txt" href="nattra_1.html#E5:58_1_1_6_1"><i><font color="Green" title="E60">A26</font></i></a>, <a class="txt" href="nattra_1.html#E7:58_1_1_6_1"><i><font color="Green" title="E62">A28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:58_1_1_6_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #)
 <b>by </b><i><a class="txt" href="nattra_1.html#E10:58_1_1"><i><font color="Green" title="E53">A14</font></i></a>, <a class="txt" href="nattra_1.html#E3:58_1_1_6_1"><i><font color="Green" title="E59">A25</font></i></a>, <a class="txt" href="nattra_1.html#E5:58_1_1_6_1"><i><font color="Green" title="E60">A26</font></i></a>, <a class="txt" href="nattra_1.html#E7:58_1_1_6_1"><i><font color="Green" title="E62">A28</font></i></a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:58_1_1_6"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #)  st the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/>( the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> &amp; the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #);<br/>

<b>assume </b><a NAME="E1:58_1_1_6_2"/>
the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font>
 ;<br/>

<a NAME="E2:58_1_1_6_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">m</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E1:58_1_1_6"><i><font color="Green" title="E58">A23</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">F</font>, <font color="Maroon" title="c11">F1</font>, <font color="Maroon" title="c12">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c13">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c11">F1</font>, <font color="Maroon" title="c14">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c11">F1</font>,<font color="Maroon" title="c12">F2</font><b> such that </b><br/><a NAME="E4:58_1_1_6_2"/><i><font color="Green" title="E59">A29</font></i>: 
( <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c11">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c13">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">F1</font>,<font color="Maroon" title="c12">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c14">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">m</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c12">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c13">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E11:58_1_1"><i><font color="Green" title="E54">A15</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">F1'</font>, <font color="Maroon" title="c16">F2'</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c17">t'</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c15">F1'</font>,<font color="Maroon" title="c16">F2'</font><b> such that </b><br/><a NAME="E6:58_1_1_6_2"/><i><font color="Green" title="E60">A30</font></i>: 
( <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F1'</font>,<font color="Maroon" title="c16">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">t'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c15">F1'</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c16">F2'</font> )
 <b>by </b><i><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></i>;<br/>
<a NAME="E7:58_1_1_6_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c11">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F1'</font>,<font color="Maroon" title="c16">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E4:58_1_1_6_2"><i><font color="Green" title="E59">A29</font></i></a>, <a class="txt" href="nattra_1.html#E6:58_1_1_6_2"><i><font color="Green" title="E60">A30</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E8:58_1_1_6_2"/><b>then </b><i><font color="Green" title="E61">A31</font></i>: 
( <font color="Maroon" title="c10">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c15">F1'</font> &amp; <font color="Maroon" title="c11">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c16">F2'</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">F1'</font>, <font color="Maroon" title="c19">F2'</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c20">t'</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c18">F1'</font>,<font color="Maroon" title="c19">F2'</font><b> such that </b><br/><a NAME="E10:58_1_1_6_2"/><i><font color="Green" title="E62">A32</font></i>: 
( <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">F1'</font>,<font color="Maroon" title="c19">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c20">t'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c18">F1'</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c19">F2'</font> )
 <b>by </b><i><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></i>;<br/>
<a NAME="E11:58_1_1_6_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">F1</font>,<font color="Maroon" title="c12">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">F1'</font>,<font color="Maroon" title="c19">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E4:58_1_1_6_2"><i><font color="Green" title="E59">A29</font></i></a>, <a class="txt" href="nattra_1.html#E10:58_1_1_6_2"><i><font color="Green" title="E62">A32</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E12:58_1_1_6_2"/><b>then </b>
( <font color="Maroon" title="c11">F1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c18">F1'</font> &amp; <font color="Maroon" title="c12">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c19">F2'</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E13:58_1_1_6_2"/><b>then </b>
<font color="Maroon" title="c10">F</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c12">F2</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E6:58_1_1_6_2"><i><font color="Green" title="E60">A30</font></i></a>, <a class="txt" href="nattra_1.html#E8:58_1_1_6_2"><i><font color="Green" title="E61">A31</font></i></a>, <a class="txt" href="nattra_1.html#E10:58_1_1_6_2"><i><font color="Green" title="E62">A32</font></i></a>, <a class="ref" href="nattra_1.html#T25" target="_self" title="NATTRA_1:th.25">Th25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c21">x</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c12">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c13">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <b>by </b><i><a class="ref" href="nattra_1.html#T35" target="_self" title="NATTRA_1:th.35">Th35</a></i>;<br/>
<b>thus </b>the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_1"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="nattra_1.html#E4:58_1_1_6_2"><i><font color="Green" title="E59">A29</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c12">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c10">F</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c11">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c11">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c13">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_1"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="nattra_1.html#E4:58_1_1_6_2"><i><font color="Green" title="E59">A29</font></i></a></i>
;<br/>

<b>thus </b>the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E6:58_1_1"><i><font color="Green" title="E50">A6</font></i></a>, <a class="txt" href="nattra_1.html#E4:58_1_1_6_2"><i><font color="Green" title="E59">A29</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c12">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c12">F2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">F1</font>,<font color="Maroon" title="c12">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">F1</font>,<font color="Maroon" title="c12">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c14">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E6:58_1_1"><i><font color="Green" title="E50">A6</font></i></a>, <a class="txt" href="nattra_1.html#E4:58_1_1_6_2"><i><font color="Green" title="E59">A29</font></i></a></i>
;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:58_1_1_6"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #)  st the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> &amp; the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> holds <br/>the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">h</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">h</font>,<font color="Olive" title="b2">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<font color="Olive" title="b1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font>, <font color="Maroon" title="c10">h</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c11">f'</font> = <font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c12">g'</font> = <font color="Maroon" title="c9">g</font>, <font color="Maroon" title="c13">h'</font> = <font color="Maroon" title="c10">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ;<br/>
<b>assume </b><a NAME="E2:58_1_1_6_3"/><i><font color="Green" title="E59">A33</font></i>: 
( the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> &amp; the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> )
 ;<br/>

<a NAME="E3:58_1_1_6_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">g'</font>,<font color="Maroon" title="c11">f'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">m</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E1:58_1_1_6"><i><font color="Green" title="E58">A23</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">F1</font>, <font color="Maroon" title="c15">F1'</font>, <font color="Maroon" title="c16">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c17">t1</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c15">F1'</font>, <font color="Maroon" title="c18">t2</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c15">F1'</font>,<font color="Maroon" title="c16">F2</font><b> such that </b><br/><a NAME="E5:58_1_1_6_3"/><i><font color="Green" title="E60">A34</font></i>: 
( <font color="Maroon" title="c11">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c15">F1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F1'</font>,<font color="Maroon" title="c16">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c18">t2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">m</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">g'</font>,<font color="Maroon" title="c11">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c16">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">t2</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c17">t1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E11:58_1_1"><i><font color="Green" title="E54">A15</font></i></a></i>;<br/>
<a NAME="E6:58_1_1_6_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">h'</font>,<font color="Maroon" title="c12">g'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">m</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E1:58_1_1_6"><i><font color="Green" title="E58">A23</font></i></a>, <a class="txt" href="nattra_1.html#E2:58_1_1_6_3"><i><font color="Green" title="E59">A33</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">F2'</font>, <font color="Maroon" title="c20">F3</font>, <font color="Maroon" title="c21">F3'</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c22">t2'</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c19">F2'</font>,<font color="Maroon" title="c20">F3</font>, <font color="Maroon" title="c23">t3</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c20">F3</font>,<font color="Maroon" title="c21">F3'</font><b> such that </b><br/><a NAME="E8:58_1_1_6_3"/><i><font color="Green" title="E61">A35</font></i>: 
( <font color="Maroon" title="c12">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F2'</font>,<font color="Maroon" title="c20">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c22">t2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">h'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">F3</font>,<font color="Maroon" title="c21">F3'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c23">t3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c5">m</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">h'</font>,<font color="Maroon" title="c12">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F2'</font>,<font color="Maroon" title="c21">F3'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">t3</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c22">t2'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E11:58_1_1"><i><font color="Green" title="E54">A15</font></i></a></i>;<br/>
<a NAME="E9:58_1_1_6_3"/><i><font color="Green" title="E62">A36</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F2'</font>,<font color="Maroon" title="c20">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F1'</font>,<font color="Maroon" title="c16">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E5:58_1_1_6_3"><i><font color="Green" title="E60">A34</font></i></a>, <a class="txt" href="nattra_1.html#E8:58_1_1_6_3"><i><font color="Green" title="E61">A35</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E10:58_1_1_6_3"/><b>then </b><i><font color="Green" title="E63">A37</font></i>: 
<font color="Maroon" title="c16">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c20">F3</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E11:58_1_1_6_3"/><i><font color="Green" title="E64">A38</font></i>: 
<font color="Maroon" title="c12">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F1'</font>,<font color="Maroon" title="c20">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c18">t2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E5:58_1_1_6_3"><i><font color="Green" title="E60">A34</font></i></a>, <a class="txt" href="nattra_1.html#E9:58_1_1_6_3"><i><font color="Green" title="E62">A36</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E12:58_1_1_6_3"/><i><font color="Green" title="E65">A39</font></i>: 
<font color="Maroon" title="c14">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c15">F1'</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E5:58_1_1_6_3"><i><font color="Green" title="E60">A34</font></i></a>, <a class="ref" href="nattra_1.html#T35" target="_self" title="NATTRA_1:th.35">Th35</a></i>;<br/>
<a NAME="E13:58_1_1_6_3"/><i><font color="Green" title="E66">A40</font></i>: 
<font color="Maroon" title="c15">F1'</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c20">F3</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E5:58_1_1_6_3"><i><font color="Green" title="E60">A34</font></i></a>, <a class="txt" href="nattra_1.html#E10:58_1_1_6_3"><i><font color="Green" title="E63">A37</font></i></a>, <a class="ref" href="nattra_1.html#T35" target="_self" title="NATTRA_1:th.35">Th35</a></i>;<br/>
<a NAME="E14:58_1_1_6_3"/><i><font color="Green" title="E67">A41</font></i>: 
<font color="Maroon" title="c20">F3</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c21">F3'</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E8:58_1_1_6_3"><i><font color="Green" title="E61">A35</font></i></a>, <a class="ref" href="nattra_1.html#T35" target="_self" title="NATTRA_1:th.35">Th35</a></i>;<br/>
<a NAME="E15:58_1_1_6_3"/><i><font color="Green" title="E68">A42</font></i>: 
<font color="Maroon" title="c14">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c20">F3</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E12:58_1_1_6_3"><i><font color="Green" title="E65">A39</font></i></a>, <a class="txt" href="nattra_1.html#E13:58_1_1_6_3"><i><font color="Green" title="E66">A40</font></i></a>, <a class="ref" href="nattra_1.html#T25" target="_self" title="NATTRA_1:th.25">Th25</a></i>;<br/>
<a NAME="E16:58_1_1_6_3"/><i><font color="Green" title="E69">A43</font></i>: 
<font color="Maroon" title="c15">F1'</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c21">F3'</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E13:58_1_1_6_3"><i><font color="Green" title="E66">A40</font></i></a>, <a class="txt" href="nattra_1.html#E14:58_1_1_6_3"><i><font color="Green" title="E67">A41</font></i></a>, <a class="ref" href="nattra_1.html#T25" target="_self" title="NATTRA_1:th.25">Th25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">t2</font> = <font color="Maroon" title="c18">t2</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c15">F1'</font>,<font color="Maroon" title="c20">F3</font> <b>by </b><i><a class="txt" href="nattra_1.html#E9:58_1_1_6_3"><i><font color="Green" title="E62">A36</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>thus </b>the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">h</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c21">F3'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">t3</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <span class="p3">(<span class="default"><font color="Maroon" title="c24">t2</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c17">t1</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="nattra_1.html#E15:58_1_1"><i><font color="Green" title="E57">A18</font></i></a>, <a class="txt" href="nattra_1.html#E5:58_1_1_6_3"><i><font color="Green" title="E60">A34</font></i></a>, <a class="txt" href="nattra_1.html#E8:58_1_1_6_3"><i><font color="Green" title="E61">A35</font></i></a>, <a class="txt" href="nattra_1.html#E10:58_1_1_6_3"><i><font color="Green" title="E63">A37</font></i></a>, <a class="txt" href="nattra_1.html#E14:58_1_1_6_3"><i><font color="Green" title="E67">A41</font></i></a>, <a class="txt" href="nattra_1.html#E15:58_1_1_6_3"><i><font color="Green" title="E68">A42</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1</font>,<font color="Maroon" title="c21">F3'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">t3</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c24">t2</font></span>)</span> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c17">t1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="nattra_1.html#E12:58_1_1_6_3"><i><font color="Green" title="E65">A39</font></i></a>, <a class="txt" href="nattra_1.html#E13:58_1_1_6_3"><i><font color="Green" title="E66">A40</font></i></a>, <a class="txt" href="nattra_1.html#E14:58_1_1_6_3"><i><font color="Green" title="E67">A41</font></i></a>, <a class="ref" href="nattra_1.html#T28" target="_self" title="NATTRA_1:th.28">Th28</a></i>
<br/>.= 
<font color="Maroon" title="c5">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F1'</font>,<font color="Maroon" title="c21">F3'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Maroon" title="c23">t3</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c24">t2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c11">f'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="nattra_1.html#E15:58_1_1"><i><font color="Green" title="E57">A18</font></i></a>, <a class="txt" href="nattra_1.html#E5:58_1_1_6_3"><i><font color="Green" title="E60">A34</font></i></a>, <a class="txt" href="nattra_1.html#E12:58_1_1_6_3"><i><font color="Green" title="E65">A39</font></i></a>, <a class="txt" href="nattra_1.html#E16:58_1_1_6_3"><i><font color="Green" title="E69">A43</font></i></a></i>
<br/>.= 
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">h</font>,<font color="Maroon" title="c9">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="nattra_1.html#E15:58_1_1"><i><font color="Green" title="E57">A18</font></i></a>, <a class="txt" href="nattra_1.html#E8:58_1_1_6_3"><i><font color="Green" title="E61">A35</font></i></a>, <a class="txt" href="nattra_1.html#E11:58_1_1_6_3"><i><font color="Green" title="E64">A38</font></i></a>, <a class="txt" href="nattra_1.html#E13:58_1_1_6_3"><i><font color="Green" title="E66">A40</font></i></a>, <a class="txt" href="nattra_1.html#E14:58_1_1_6_3"><i><font color="Green" title="E67">A41</font></i></a></i>
;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon" title="c8">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #);<br/><b>reconsider </b><font color="Maroon" title="c9">F</font> = <font color="Maroon" title="c8">b</font> as    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <b>by </b><i><a class="ref" href="cat_2.html#D2" title="CAT_2:def.2">CAT_2:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">s</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c9">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <b>by </b><i><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></i>;<br/><a NAME="E6:58_1_1_6"/><i><font color="Green" title="E59">A44</font></i>: 
the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c9">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E14:58_1_1"><i><font color="Green" title="E56">A17</font></i></a></i>;<br/><b>hence </b>the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_1"><i><font color="Green" title="E48">A3</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c8">b</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/><br/><b>thus </b>the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E6:58_1_1"><i><font color="Green" title="E50">A6</font></i></a>, <a class="txt" href="nattra_1.html#E6:58_1_1_6"><i><font color="Green" title="E59">A44</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c8">b</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/><b>thus </b><a NAME="E9:58_1_1_6"/>
for <font color="Olive" title="b1">f</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #)  st the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> holds <br/>the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<font color="Olive" title="b1">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_6_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">f</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c12">f'</font> = <font color="Maroon" title="c11">f</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ;<br/>
<b>consider </b><font color="Maroon" title="c13">F1</font>, <font color="Maroon" title="c14">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c15">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font><b> such that </b><br/><a NAME="E3:58_1_1_6_6"/><i><font color="Green" title="E60">A45</font></i>: 
<font color="Maroon" title="c12">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E4:58_1_1_6_6"/>
<font color="Maroon" title="c13">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c14">F2</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></i>;<br/>
<b>assume </b><a NAME="E5:58_1_1_6_6"/><i><font color="Green" title="E61">A46</font></i>: 
the <a href="cat_1.html#U4" title="CAT_1:sel.4">Cod</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 ;<br/>

<i><font color="Green" title="E62">A47</font></i>: <font color="Maroon" title="c14">F2</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f'</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_1_6_6"><i><font color="Green" title="E60">A45</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c9">F</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E6:58_1_1"><i><font color="Green" title="E50">A6</font></i></a>, <a class="txt" href="nattra_1.html#E5:58_1_1_6_6"><i><font color="Green" title="E61">A46</font></i></a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">t</font> = <font color="Maroon" title="c15">t</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c9">F</font> ;<br/>
<a NAME="E8:58_1_1_6_6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">f'</font>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c16">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]
 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_1_6_6"><i><font color="Green" title="E60">A45</font></i></a>, <a class="txt" href="nattra_1.html#E6:58_1_1_6_6"><i><font color="Green" title="E62">A47</font></i></a></i>;<br/>
<a NAME="E9:58_1_1_6_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">s</font>,<font color="Maroon" title="c12">f'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">m</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E10:58_1_1"><i><font color="Green" title="E53">A14</font></i></a></i>;<br/>
<a NAME="E10:58_1_1_6_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">F</font></span>)</span>,<font color="Maroon" title="c12">f'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">m</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E14:58_1_1"><i><font color="Green" title="E56">A17</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">F'</font>, <font color="Maroon" title="c18">F1'</font>, <font color="Maroon" title="c19">F2'</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c20">t'</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c17">F'</font>,<font color="Maroon" title="c18">F1'</font>, <font color="Maroon" title="c21">t1'</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c18">F1'</font>,<font color="Maroon" title="c19">F2'</font><b> such that </b><br/><a NAME="E12:58_1_1_6_6"/><i><font color="Green" title="E63">A48</font></i>: 
( <font color="Maroon" title="c12">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F'</font>,<font color="Maroon" title="c18">F1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c20">t'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">F1'</font>,<font color="Maroon" title="c19">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c21">t1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>and </b><br/><a NAME="E13:58_1_1_6_6"/><i><font color="Green" title="E64">A49</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">F</font></span>)</span>,<font color="Maroon" title="c12">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F'</font>,<font color="Maroon" title="c19">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">t1'</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c20">t'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E11:58_1_1"><i><font color="Green" title="E54">A15</font></i></a></i>;<br/>
<a NAME="E14:58_1_1_6_6"/><i><font color="Green" title="E65">A50</font></i>: 
<font color="Maroon" title="c17">F'</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c18">F1'</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E12:58_1_1_6_6"><i><font color="Green" title="E63">A48</font></i></a>, <a class="ref" href="nattra_1.html#T35" target="_self" title="NATTRA_1:th.35">Th35</a></i>;<br/>
<a NAME="E15:58_1_1_6_6"/>
<font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c9">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E14:58_1_1"><i><font color="Green" title="E56">A17</font></i></a></i>;<br/>
<a NAME="E16:58_1_1_6_6"/><b>then </b><i><font color="Green" title="E66">A51</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">F1'</font>,<font color="Maroon" title="c19">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c21">t1'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c9">F</font> )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E12:58_1_1_6_6"><i><font color="Green" title="E63">A48</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E17:58_1_1_6_6"/><b>then </b>
( <font color="Maroon" title="c18">F1'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">F</font> &amp; <font color="Maroon" title="c19">F2'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">F</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E18:58_1_1_6_6"/>
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span>,<font color="Maroon" title="c11">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E12:58_1_1_6_6"><i><font color="Green" title="E63">A48</font></i></a>, <a class="txt" href="nattra_1.html#E13:58_1_1_6_6"><i><font color="Green" title="E64">A49</font></i></a>, <a class="txt" href="nattra_1.html#E14:58_1_1_6_6"><i><font color="Green" title="E65">A50</font></i></a>, <a class="txt" href="nattra_1.html#E16:58_1_1_6_6"><i><font color="Green" title="E66">A51</font></i></a>, <a class="ref" href="nattra_1.html#T26" target="_self" title="NATTRA_1:th.26">Th26</a></i>;<br/>


</div><b>end;</b></div><b>let </b><font color="Maroon" title="c11">g</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #);<br/><b>reconsider </b><font color="Maroon" title="c12">g'</font> = <font color="Maroon" title="c11">g</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ;<br/><b>consider </b><font color="Maroon" title="c13">F1</font>, <font color="Maroon" title="c14">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c15">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font><b> such that </b><br/><a NAME="E12:58_1_1_6"/><i><font color="Green" title="E60">A52</font></i>: 
<font color="Maroon" title="c12">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E13:58_1_1_6"/>
<font color="Maroon" title="c13">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c14">F2</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></i>;<br/><b>assume </b><a NAME="E14:58_1_1_6"/><i><font color="Green" title="E61">A53</font></i>: 
the <a href="cat_1.html#U3" title="CAT_1:sel.3">Dom</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 ;<br/><i><font color="Green" title="E62">A54</font></i>: <font color="Maroon" title="c13">F1</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">g'</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E12:58_1_1_6"><i><font color="Green" title="E60">A52</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c9">F</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_1"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="nattra_1.html#E14:58_1_1_6"><i><font color="Green" title="E61">A53</font></i></a></i>
;<br/><b>then reconsider </b><font color="Maroon" title="c16">t</font> = <font color="Maroon" title="c15">t</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c14">F2</font> ;<br/><a NAME="E17:58_1_1_6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">g'</font>,<font color="Maroon" title="c10">s</font>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">t</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <span class="p3">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c9">F</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]
 <b>by </b><i><a class="txt" href="nattra_1.html#E12:58_1_1_6"><i><font color="Green" title="E60">A52</font></i></a>, <a class="txt" href="nattra_1.html#E15:58_1_1_6"><i><font color="Green" title="E62">A54</font></i></a></i>;<br/><a NAME="E18:58_1_1_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">g'</font>,<font color="Maroon" title="c10">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E10:58_1_1"><i><font color="Green" title="E53">A14</font></i></a></i>;<br/><a NAME="E19:58_1_1_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">g'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">m</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E14:58_1_1"><i><font color="Green" title="E56">A17</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">F'</font>, <font color="Maroon" title="c18">F1'</font>, <font color="Maroon" title="c19">F2'</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c20">t'</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c17">F'</font>,<font color="Maroon" title="c18">F1'</font>, <font color="Maroon" title="c21">t1'</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c18">F1'</font>,<font color="Maroon" title="c19">F2'</font><b> such that </b><br/><a NAME="E21:58_1_1_6"/><i><font color="Green" title="E63">A55</font></i>: 
( <font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F'</font>,<font color="Maroon" title="c18">F1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c20">t'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">F1'</font>,<font color="Maroon" title="c19">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c21">t1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>and </b><br/><a NAME="E22:58_1_1_6"/><i><font color="Green" title="E64">A56</font></i>: 
<font color="Maroon" title="c5">m</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">g'</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F'</font>,<font color="Maroon" title="c19">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c21">t1'</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c20">t'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E11:58_1_1"><i><font color="Green" title="E54">A15</font></i></a></i>;<br/><a NAME="E23:58_1_1_6"/><i><font color="Green" title="E65">A57</font></i>: 
<font color="Maroon" title="c18">F1'</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c19">F2'</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E21:58_1_1_6"><i><font color="Green" title="E63">A55</font></i></a>, <a class="ref" href="nattra_1.html#T35" target="_self" title="NATTRA_1:th.35">Th35</a></i>;<br/><a NAME="E24:58_1_1_6"/>
<font color="Maroon" title="c6">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c9">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E14:58_1_1"><i><font color="Green" title="E56">A17</font></i></a></i>;<br/><a NAME="E25:58_1_1_6"/><b>then </b><i><font color="Green" title="E66">A58</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F'</font>,<font color="Maroon" title="c18">F1'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c9">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c20">t'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c9">F</font> )
 <b>by </b><i><a class="txt" href="nattra_1.html#E21:58_1_1_6"><i><font color="Green" title="E63">A55</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E26:58_1_1_6"/><b>then </b>
( <font color="Maroon" title="c18">F1'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">F</font> &amp; <font color="Maroon" title="c17">F'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">F</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E27:58_1_1_6"/>
the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">g</font>,<span class="p2">(<span class="default">the <a href="cat_1.html#U6" title="CAT_1:sel.6">Id</a> of <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E21:58_1_1_6"><i><font color="Green" title="E63">A55</font></i></a>, <a class="txt" href="nattra_1.html#E22:58_1_1_6"><i><font color="Green" title="E64">A56</font></i></a>, <a class="txt" href="nattra_1.html#E23:58_1_1_6"><i><font color="Green" title="E65">A57</font></i></a>, <a class="txt" href="nattra_1.html#E25:58_1_1_6"><i><font color="Green" title="E66">A58</font></i></a>, <a class="ref" href="nattra_1.html#T26" target="_self" title="NATTRA_1:th.26">Th26</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">C</font> =  <a href="cat_1.html#G1" title="CAT_1:aggr.1">CatStr</a>(# <span class="p1">(<span class="default"><a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">m</font>,<font color="Maroon" title="c6">i</font> #) as  <a href="cat_1.html#V1" title="CAT_1:attr.1">strict</a> <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a> <b>by </b><i><a class="ref" href="cat_1.html#D8" title="CAT_1:def.8">CAT_1:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">C</font>
;<br/>

<b>thus </b><a NAME="E18:58_1_1"/>
( the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <font color="Maroon" title="c8">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; the <a href="cat_1.html#U2" title="CAT_1:sel.2">Morphisms</a> of <font color="Maroon" title="c8">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nattra_1.html#K11" title="NATTRA_1:func.11">NatTrans</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> )
 ;<br/>

<b>thus </b><a NAME="E19:58_1_1"/><i><font color="Green" title="E58">A59</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font> holds <br/> (  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 <b>by </b><i><a class="txt" href="nattra_1.html#E3:58_1_1"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="nattra_1.html#E6:58_1_1"><i><font color="Green" title="E50">A6</font></i></a></i>;<br/>

<b>thus </b><a NAME="E20:58_1_1"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>  st  <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c8">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>;<br/>

<b>consider </b><font color="Maroon" title="c11">F1</font>, <font color="Maroon" title="c12">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c13">t</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c11">F1</font>,<font color="Maroon" title="c12">F2</font><b> such that </b><br/><a NAME="E2:58_1_1_7"/><i><font color="Green" title="E59">A60</font></i>: 
( <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">F1</font>,<font color="Maroon" title="c12">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c13">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">F1</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c12">F2</font> )
 <b>by </b><i><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">F1'</font>, <font color="Maroon" title="c15">F2'</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c16">t'</font> being    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c14">F1'</font>,<font color="Maroon" title="c15">F2'</font><b> such that </b><br/><a NAME="E4:58_1_1_7"/><i><font color="Green" title="E60">A61</font></i>: 
( <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1'</font>,<font color="Maroon" title="c15">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c16">t'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">F1'</font> <a href="nattra_1.html#R2" title="NATTRA_1:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c15">F2'</font> )
 <b>by </b><i><a class="ref" href="nattra_1.html#D16" target="_self" title="NATTRA_1:def.16">Def16</a></i>;<br/>
<b>assume </b><a NAME="E5:58_1_1_7"/><i><font color="Green" title="E61">A62</font></i>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">dom</a> <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c9">f</font>
 ;<br/>

<i><font color="Green" title="E62">A63</font></i>: <font color="Maroon" title="c12">F2</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">F1</font>,<font color="Maroon" title="c12">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">F1</font>,<font color="Maroon" title="c12">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c13">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
 <a href="cat_1.html#K3" title="CAT_1:func.3">cod</a> <font color="Maroon" title="c9">f</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E19:58_1_1"><i><font color="Green" title="E58">A59</font></i></a>, <a class="txt" href="nattra_1.html#E2:58_1_1_7"><i><font color="Green" title="E59">A60</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1'</font>,<font color="Maroon" title="c15">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c16">t'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="txt" href="nattra_1.html#E19:58_1_1"><i><font color="Green" title="E58">A59</font></i></a>, <a class="txt" href="nattra_1.html#E4:58_1_1_7"><i><font color="Green" title="E60">A61</font></i></a>, <a class="txt" href="nattra_1.html#E5:58_1_1_7"><i><font color="Green" title="E61">A62</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F1'</font>,<font color="Maroon" title="c15">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon" title="c14">F1'</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">t'</font> = <font color="Maroon" title="c16">t'</font> as    <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Maroon" title="c12">F2</font>,<font color="Maroon" title="c15">F2'</font> ;<br/>
<a NAME="E8:58_1_1_7"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">F1</font>,<font color="Maroon" title="c15">F2'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">t'</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Maroon" title="c13">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>]
 
<b>by </b><i><a class="txt" href="nattra_1.html#E2:58_1_1_7"><i><font color="Green" title="E59">A60</font></i></a>, <a class="txt" href="nattra_1.html#E4:58_1_1_7"><i><font color="Green" title="E60">A61</font></i></a>, <a class="txt" href="nattra_1.html#E6:58_1_1_7"><i><font color="Green" title="E62">A63</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:58_1_1_7"/>
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c8">C</font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E10:58_1_1"><i><font color="Green" title="E53">A14</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E21:58_1_1"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>  st <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c8">C</font> holds <br/> ex <font color="Olive" title="b3">F</font>, <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b6">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font> ex <font color="Olive" title="b7">t1</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b4">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c8">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b6">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">f</font>, <font color="Maroon" title="c10">g</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">C</font>;<br/>

<b>assume </b><a NAME="E1:58_1_1_8"/>
<span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c8">C</font>
 ;<br/>

<a NAME="E2:58_1_1_8"/><b>then </b>
 ex <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b4">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font> ex <font color="Olive" title="b5">t1</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font> st <br/>( <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b5">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c8">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b4">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E11:58_1_1"><i><font color="Green" title="E54">A15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:58_1_1_8"/>
 ex <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> ex <font color="Olive" title="b4">t</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font> ex <font color="Olive" title="b5">t1</font> being   <a href="nattra_1.html#M2" title="NATTRA_1:mode.2">natural_transformation</a> of <font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font> st <br/>( <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b5">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; the <a href="cat_1.html#U5" title="CAT_1:sel.5">Comp</a> of <font color="Maroon" title="c8">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="cat_2.html#K13" title="CAT_2:func.13">[</a><span class="default"><font color="Maroon" title="c10">g</font>,<font color="Maroon" title="c9">f</font></span><a href="cat_2.html#K13" title="CAT_2:func.13">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">t1</font> <a href="nattra_1.html#K8" title="NATTRA_1:func.8">`*`</a> <font color="Olive" title="b4">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ;<br/>


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

<b>let </b><font color="Maroon" title="c9">a</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c8">C</font>;<br/><b>let </b><font color="Maroon" title="c10">F</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>;<br/>



<b>assume </b><a NAME="E22:58_1_1"/><i><font color="Green" title="E59">A64</font></i>: 
<font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c11">F'</font> = <font color="Maroon" title="c10">F</font> as    <a href="cat_2.html#M2" title="CAT_2:mode.2">Element</a> of  <a href="cat_2.html#K7" title="CAT_2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <b>by </b><i><a class="ref" href="cat_2.html#D2" title="CAT_2:def.2">CAT_2:def 2</a></i>;<br/>
<b>thus </b><a NAME="E24:58_1_1"/>
 <a href="cat_1.html#K10" title="CAT_1:func.10">id</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">F</font>,<font color="Maroon" title="c10">F</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K7" title="NATTRA_1:func.7">id</a> <font color="Maroon" title="c10">F</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E22:58_1_1"><i><font color="Green" title="E59">A64</font></i></a>, <a class="txt" href="nattra_1.html#E14:58_1_1"><i><font color="Green" title="E56">A17</font></i></a></i>;<br/>


</div>
