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<b>let </b><font color="Maroon">A</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">o</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">o</font> as   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <b>by </b><i><a class="ref" href="cat_1.html#T32">CAT_1:32</a></i>;<br/>
<b>take </b><font color="Maroon">F</font> = <font color="Maroon">a</font> <a href="isocat_2.html#K2" target="_self">|-&gt;</a> <font color="Maroon">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="isocat_1.html#D4">ISOCAT_1:def 4</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:13_1"/><i><font color="Green">E16</font></i>: 
the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="nattra_1.html#K11">NatTrans</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/><b>assume </b><a NAME="E2:13_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">A</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">F1</font> being    <a href="cat_1.html#M2">Functor</a> of  <a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font>,<font color="Maroon">A</font>, <font color="Maroon">F2</font> being    <a href="cat_1.html#M2">Functor</a> of  <a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font>,<font color="Maroon">A</font>, <font color="Maroon">t</font> being    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font><b> such that </b><br/><a NAME="E4:13_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font> )
 <b>by </b><i>, <a class="ref" href="nattra_1.html#D16">NATTRA_1:def 16</a></i>;<br/><b>assume </b><a NAME="E5:13_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">A</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">F1'</font> being    <a href="cat_1.html#M2">Functor</a> of  <a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font>,<font color="Maroon">A</font>, <font color="Maroon">F2'</font> being    <a href="cat_1.html#M2">Functor</a> of  <a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font>,<font color="Maroon">A</font>, <font color="Maroon">t'</font> being    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1'</font>,<font color="Maroon">F2'</font><b> such that </b><br/><a NAME="E7:13_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1'</font>,<font color="Maroon">F2'</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">F1'</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2'</font> )
 <b>by </b><i>, <a class="ref" href="nattra_1.html#D16">NATTRA_1:def 16</a></i>;<br/><a NAME="E8:13_1"/><i><font color="Green">E33</font></i>: 
( the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">o</font></span><a href="tarski.html#K1">}</a></span> &amp; the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">t</font>, <font color="Maroon">s'</font> = <font color="Maroon">t'</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font> ;<br/><b>reconsider </b><font color="Maroon">G1</font> = <font color="Maroon">F1</font>, <font color="Maroon">G1'</font> = <font color="Maroon">F1'</font>, <font color="Maroon">G2</font> = <font color="Maroon">F2</font>, <font color="Maroon">G2'</font> = <font color="Maroon">F2'</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/><a NAME="E11:13_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">F1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">F1'</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">F2'</font> )
 <b>by </b><i>, , <a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a></i>;<br/><b>assume </b><a NAME="E12:13_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/><b>then </b><i><font color="Green">E51</font></i>: <font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">t'</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font>
<b>by </b><i>, </i>
;<br/><b>then </b><i><font color="Green">E54</font></i>: <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">t'</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E1:13_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
<br/>.= 
<font color="Maroon">s'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E1:13_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
;<br/><a NAME="E15:13_1"/><i><font color="Green">E55</font></i>: 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T35">CAT_1:35</a></i>;<br/><a NAME="E16:13_1"/>
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1'</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2'</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="isocat_2.html#E1:13_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/><a NAME="E17:13_1"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1'</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2'</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="txt" href="isocat_2.html#E4:13_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="cat_1.html#T24">CAT_1:24</a></i>;<br/><b>then </b><i><font color="Green">E58</font></i>: <font color="Maroon">G1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span> = 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">F1'</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</a></i>
<br/>.= 
<font color="Maroon">G1'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</a></i>
;<br/><font color="Maroon">G2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span> = 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">F2'</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#T108">CAT_1:108</a></i>
<br/>.= 
<font color="Maroon">G2'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</a></i>
;<br/><a NAME="E20:13_1"/><b>then </b>
( <font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F1'</font> &amp; <font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F2'</font> )
 <b>by </b><i><a class="ref" href="isocat_2.html#T2" target="_self">Th2</a>, , </i>;<br/><b>hence </b><a NAME="E21:13_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="isocat_2.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:13"/>
<font color="Maroon">F</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="isocat_1.html#D3">ISOCAT_1:def 3</a><br/>

<b>thus </b><a NAME="E4:13"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E5:13"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> ;<br/>
<i><font color="Green">E59</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i/>
;<br/>
<i><font color="Green">E60</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E9:13"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E10:13"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:13"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E12:13"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F1</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>, <font color="Maroon">F2</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font> <b>by </b><i>, , , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E14:13"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">g</font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> holds <br/> ( <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/>

<a NAME="E1:13_4"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T35">CAT_1:35</a></i>;<br/>
<a NAME="E2:13_4"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:13_4"/>
( <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/><b>take </b><font color="Maroon">d</font> =  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font>;<br/><b>thus </b><a NAME="E1:13_5"/>
<font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/><b>thus </b><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/><b>thus </b><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">h</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>, <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/><b>assume </b><a NAME="E1:13_7"/>
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">h</font>
 ;<br/><a NAME="E2:13_7"/><i><font color="Green">E68</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="cat_1.html#T56">CAT_1:56</a></i>;<br/><b>thus </b><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font></span>)</span> = 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T59">CAT_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="isocat_2.html#T2" target="_self">Th2</a>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/><b>take </b><font color="Maroon">d</font> =  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>;<br/><b>thus </b><a NAME="E1:13_8"/>
<font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/><b>thus </b><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/><b>thus </b><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">g</font></span>)</span></span>)</span> = 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p3">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_10"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">h</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>, <font color="Maroon">g</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/><b>assume </b><a NAME="E1:13_10"/>
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">h</font>
 ;<br/><a NAME="E2:13_10"/><i><font color="Green">E71</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="cat_1.html#T56">CAT_1:56</a></i>;<br/><b>thus </b><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <font color="Maroon">h</font></span>)</span> = 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cat_1.html#T59">CAT_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">F1</font>, <font color="Maroon">F2</font> = <font color="Maroon">F2</font> as    <a href="cat_1.html#M2">Functor</a> of  <a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font>,<font color="Maroon">A</font> <b>by </b><i>, , <a class="ref" href="isocat_2.html#T1" target="_self">Th1</a>, <a class="ref" href="isocat_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="cat_1.html#T96">CAT_1:96</a></i>;<br/>
<a NAME="E22:13"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> holds <br/> ( <font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/>

<a NAME="E1:13_11"/>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T35">CAT_1:35</a></i>;<br/>
<a NAME="E2:13_11"/><b>then </b>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:13_11"/><b>then </b>
( <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E4:13_11"/><b>then </b>
(  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="cat_1.html#T108">CAT_1:108</a></i>;<br/>
<b>hence </b><a NAME="E5:13_11"/>
( <font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T45">CAT_1:45</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_12"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/><a NAME="E1:13_12"/>
( <font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:13_12"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="cat_1.html#T19">CAT_1:19</a></i>;<br/></div><b>end;</b></div>
<a NAME="E24:13"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">F2</font>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>;<br/>
<i><font color="Green">E77</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">f</font></span>)</span> = 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E26:13"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">f</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E27:13"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">t</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_14"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/><a NAME="E1:13_14"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T34">CAT_1:34</a></i>;<br/><a NAME="E2:13_14"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:13_14"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E4:13_14"/>
( <font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/><a NAME="E5:13_14"/><b>then </b>
<font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/><b>hence </b><a NAME="E6:13_14"/>
<font color="Maroon">t</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> is    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font>,<font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="cat_1.html#D7">CAT_1:def 7</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as    <a href="nattra_1.html#M1">transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font> <b>by </b><i>, <a class="ref" href="nattra_1.html#D3">NATTRA_1:def 3</a></i>;<br/>
<a NAME="E31:13"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span> holds  <font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_15"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/>

<a NAME="E1:13_15"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T34">CAT_1:34</a></i>;<br/>
<a NAME="E2:13_15"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="nattra_1.html#D5">NATTRA_1:def 5</a></i>
;<br/><br/>


</div><b>end;</b></div>
<div><i><font color="Green">E82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_16"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b1</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>, <font color="Maroon">b2</font> be   <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>;<br/><b>assume </b><a NAME="E1:13_16"/><i><font color="Green">E83</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>let </b><font color="Maroon">g</font> be    <a href="cat_1.html#M1">Morphism</a> of <font color="Maroon">b1</font>,<font color="Maroon">b2</font>;<br/><a NAME="E2:13_16"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><a NAME="E3:13_16"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><a NAME="E4:13_16"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="cat_1.html#T35">CAT_1:35</a></i>;<br/><a NAME="E5:13_16"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">m</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E6:13_16"/><i><font color="Green">E96</font></i>: 
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="isocat_2.html#T3" target="_self">Th3</a>, <a class="ref" href="isocat_2.html#T5" target="_self">Th5</a>, <a class="ref" href="cat_1.html#T124">CAT_1:124</a></i>;<br/><a NAME="E7:13_16"/><i><font color="Green">E97</font></i>: 
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="cat_1.html#K13">.</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="isocat_2.html#T3" target="_self">Th3</a>, <a class="ref" href="isocat_2.html#T5" target="_self">Th5</a>, <a class="ref" href="cat_1.html#T124">CAT_1:124</a></i>;<br/><i><font color="Green">E98</font></i>: <font color="Maroon">F2</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font> = 
<font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="isocat_2.html#T5" target="_self">Th5</a>, , <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><i><font color="Green">E99</font></i>: <font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font> = 
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="isocat_2.html#T5" target="_self">Th5</a>, , <a class="ref" href="nattra_1.html#D1">NATTRA_1:def 1</a></i>
<br/>.= 
 <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b2</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font></span>)</span> = 
<font color="Maroon">f</font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#T47">CAT_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="cat_1.html#T46">CAT_1:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="nattra_1.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="cat_1.html#D13">CAT_1:def 13</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E33:13"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">F1</font> <a href="nattra_1.html#R2">is_naturally_transformable_to</a> <font color="Maroon">F2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_17"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:13_17"/>
<font color="Maroon">F1</font> <a href="nattra_1.html#R1">is_transformable_to</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="ref" href="isocat_2.html#T3" target="_self">Th3</a>, <a class="ref" href="nattra_1.html#D2">NATTRA_1:def 2</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D7">NATTRA_1:def 7</a><br/>

<b>thus </b><a NAME="E2:13_17"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="nattra_1.html#M1">transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font> st <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>  holds <br/>(  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or for <font color="Olive">b<sub>4</sub></font> being   <a href="cat_1.html#M1">Morphism</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> holds  <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="nattra_1.html#K3">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="cat_1.html#K9">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nattra_1.html#K5">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="isocat_2.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as    <a href="nattra_1.html#M2">natural_transformation</a> of <font color="Maroon">F1</font>,<font color="Maroon">F2</font> <b>by </b><i><a class="ref" href="isocat_2.html#T4" target="_self">Th4</a>, <a class="ref" href="nattra_1.html#D8">NATTRA_1:def 8</a></i>;<br/>
<a NAME="E35:13"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="nattra_1.html#K11">NatTrans</a> <span class="p1">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="isocat_2.html#T5" target="_self">Th5</a>, <a class="ref" href="nattra_1.html#D16">NATTRA_1:def 16</a></i>;<br/>
<a NAME="E36:13"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K12">Functors</a> <span class="p2">(<span class="default"><a href="cat_1.html#K8">1Cat</a> <font color="Maroon">o</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nattra_1.html#D18">NATTRA_1:def 18</a></i>;<br/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F1</font>,<font color="Maroon">F2</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> = 
<font color="Maroon">t</font> <a href="nattra_1.html#K5">.</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="isocat_2.html#T5" target="_self">Th5</a>, </i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E38:13"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>


</div>
