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<b>let </b><font color="Maroon">A</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">F1</font> be    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>, <font color="Maroon">F2</font> be    <a href="cat_1.html#M2">Functor</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>,<font color="Maroon">C</font>;<br/>







<b>assume </b><a NAME="E1:30"/><i><font color="Green">E16</font></i>: 
 <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a>  <a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">fg</font> be   <a href="cat_1.html#NM2">Morphism</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="cat_2.html#K11">:]</a></span>;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font>, <font color="Maroon">g</font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E2:30_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">fg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="cat_2.html#K13">]</a></span>
 <b>by </b><i><a class="ref" href="cat_2.html#T37">CAT_2:37</a></i>;<br/><a NAME="E3:30_1"/><i><font color="Green">E32</font></i>: 
(  <a href="cat_1.html#K2">dom</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> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font> &amp;  <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> <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> )
 <b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>;<br/><i><font color="Green">E33</font></i>: <font color="Maroon">fg</font> = 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<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> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">g</font></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<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> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="cat_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="cat_2.html#K13">[</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>,<font color="Maroon">g</font></span><a href="cat_2.html#K13">]</a></span> <a href="cat_1.html#K4">*</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f</font>,<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">g</font></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i>, <a class="ref" href="cat_2.html#T39">CAT_2:39</a></i>
;<br/><i><font color="Green">E50</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"><span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <span class="p4">(<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#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="isocat_2.html#K4" target="_self">?-</a> <font color="Maroon">f</font></span>)</span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><a href="isocat_2.html#K5" target="_self">export</a> <font color="Maroon">F2</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="isocat_2.html#T5" target="_self">Th5</a>, <a class="ref" href="isocat_2.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</a> <span class="p4">(<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#K4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="isocat_2.html#K4" target="_self">?-</a> <font color="Maroon">f</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="isocat_2.html#T3" target="_self">Th3</a></i>
;<br/><a NAME="E6:30_1"/><b>then </b>
( <font color="Maroon">F1</font> <a href="isocat_2.html#K4" target="_self">?-</a> <font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F2</font> <a href="isocat_2.html#K4" target="_self">?-</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</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 class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</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#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</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 class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</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#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E7:30_1"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">F1</font> <a href="cat_2.html#K15">?-</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</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">F1</font> <a href="cat_2.html#K15">?-</a> <span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</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="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><i><font color="Green">E54</font></i>: <font color="Maroon">F1</font> <a href="cat_2.html#K14">.</a> <font color="Maroon">f</font>,<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> = 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="isocat_2.html#K4" target="_self">?-</a> <font color="Maroon">f</font></span>)</span> <a href="nattra_1.html#K5">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="isocat_2.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="isocat_2.html#K4" target="_self">?-</a> <font color="Maroon">f</font></span>)</span> <a href="nattra_1.html#K5">.</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i>, <a class="txt" href="isocat_2.html#E3:30_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>
<br/>.= 
<font color="Maroon">F2</font> <a href="cat_2.html#K14">.</a> <font color="Maroon">f</font>,<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>
<b>by </b><i><a class="ref" href="isocat_2.html#T2" target="_self">Th2</a></i>
;<br/><i><font color="Green">E55</font></i>: <font color="Maroon">F1</font> <a href="cat_2.html#K14">.</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>,<font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="cat_2.html#K15">?-</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="funct_2.html#K3">.</a> <font color="Maroon">g</font>
<b>by </b><i><a class="txt" href="isocat_2.html#E3:30_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="cat_2.html#T47">CAT_2:47</a></i>
<br/>.= 
<font color="Maroon">F2</font> <a href="cat_2.html#K14">.</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>,<font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="cat_2.html#T47">CAT_2:47</a></i>
;<br/><i><font color="Green">E57</font></i>:  <a href="cat_1.html#K3">cod</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f</font>,<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">g</font></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span> = 
<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span>)</span></span><a href="cat_2.html#K12">]</a></span>
<b>by </b><i><a class="ref" href="cat_2.html#T38">CAT_2:38</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span><a href="cat_2.html#K12">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span><a href="cat_2.html#K12">]</a></span>
<b>by </b><i><a class="ref" href="cat_1.html#T44">CAT_1:44</a></i>
<br/>.= 
 <a href="cat_1.html#K2">dom</a> <span class="p1"><a href="cat_2.html#K13">[</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>,<font color="Maroon">g</font></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="ref" href="cat_2.html#T38">CAT_2:38</a></i>
;<br/><b>hence </b><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">fg</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p2"><a href="cat_2.html#K13">[</a><span class="default"><span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<font color="Maroon">g</font></span><a href="cat_2.html#K13">]</a></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> <span class="p2"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <span class="p4">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span></span><a href="cat_2.html#K13">]</a></span></span>)</span>
<b>by </b><i>, <a class="txt" href="isocat_2.html#E4:30_1"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="cat_1.html#T99">CAT_1:99</a></i>
<br/>.= 
<font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">fg</font>
<b>by </b><i>, , <a class="ref" href="cat_1.html#T99">CAT_1:99</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:30"/>
<font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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