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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>take </b><font color="Maroon">F</font> = <span class="p1"><a href="cat_2.html#K19">&lt;:</a><span class="default"><span class="p2">(<span class="default"><a href="cat_2.html#K18">pr2</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_2.html#K17">pr1</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span></span><a href="cat_2.html#K19">:&gt;</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="isocat_1.html#D4" target="_self">ISOCAT_1:def 4</a><br/>

<a NAME="E1:22"/><i><font color="Green">E18</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="cat_2.html#K17">pr1</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</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> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="cat_2.html#K18">pr2</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</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> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:22_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">x11</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x12</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">x11</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">x12</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">B</font> )
 <b>and </b><br/><a NAME="E4:22_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x11</font>,<font color="Maroon">x12</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>assume </b><a NAME="E5:22_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">B</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">x21</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x22</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:22_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">x21</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font> &amp; <font color="Maroon">x22</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">B</font> )
 <b>and </b><br/><a NAME="E8:22_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x21</font>,<font color="Maroon">x22</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">f11</font> = <font color="Maroon">x11</font>, <font color="Maroon">f21</font> = <font color="Maroon">x21</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="isocat_1.html#T1" target="_self">Th1</a>, <a class="txt" href="isocat_1.html#E3:22_1"><i><font color="Green">E19</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">f12</font> = <font color="Maroon">x12</font>, <font color="Maroon">f22</font> = <font color="Maroon">x22</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">B</font> <b>by </b><i><a class="ref" href="isocat_1.html#T1" target="_self">Th1</a>, <a class="txt" href="isocat_1.html#E3:22_1"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E11:22_1"/><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f21</font>,<font color="Maroon">f22</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</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/><a NAME="E12:22_1"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f11</font>,<font color="Maroon">f12</font></span><a href="cat_2.html#K13">]</a></span> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</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>assume </b><a NAME="E13:22_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/><span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f12</font>,<font color="Maroon">f11</font></span><a href="cat_2.html#K13">]</a></span> = 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f12</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K17">pr1</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="cat_2.html#K14">.</a> <font color="Maroon">f11</font>,<font color="Maroon">f12</font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a></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_2.html#K18">pr2</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="cat_2.html#K14">.</a> <font color="Maroon">f11</font>,<font color="Maroon">f12</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K17">pr1</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="cat_2.html#K14">.</a> <font color="Maroon">f11</font>,<font color="Maroon">f12</font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="ref" href="funct_3.html#D6">FUNCT_3:def 6</a></i>
<br/>.= 
<font color="Maroon">F</font> <a href="cat_2.html#K14">.</a> <font color="Maroon">f21</font>,<font color="Maroon">f22</font>
<b>by </b><i>, <a class="txt" href="isocat_1.html#E1:22"><i><font color="Green">E18</font></i></a>, , <a class="txt" href="isocat_1.html#E7:22_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_3.html#T69">FUNCT_3:69</a>, <a class="txt" href="isocat_1.html#E4:22_1"><i><font color="Green">E22</font></i></a></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_2.html#K18">pr2</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="cat_2.html#K14">.</a> <font color="Maroon">f21</font>,<font color="Maroon">f22</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K17">pr1</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="cat_2.html#K14">.</a> <font color="Maroon">f21</font>,<font color="Maroon">f22</font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_3.html#T69">FUNCT_3:69</a>, </i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f22</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K17">pr1</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="cat_2.html#K14">.</a> <font color="Maroon">f21</font>,<font color="Maroon">f22</font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="ref" href="funct_3.html#D6">FUNCT_3:def 6</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">f22</font>,<font color="Maroon">f21</font></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a></i>
;<br/><a NAME="E15:22_1"/><b>then </b>
( <font color="Maroon">x12</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x22</font> &amp; <font color="Maroon">x11</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x21</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E16:22_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="txt" href="isocat_1.html#E1:22"><i><font color="Green">E18</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:22"/>
<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" target="_self">ISOCAT_1:def 3</a><br/>

<b>thus </b><a NAME="E4:22"/>
 <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 <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="cat_2.html#K11">:]</a></span>
 ; <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="E5:22"/>
<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"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">B</font>,<font color="Maroon">A</font></span><a href="cat_2.html#K11">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:22"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">B</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">A</font> )
 <b>and </b><br/><a NAME="E8:22"/><i><font color="Green">E60</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"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">B</font> <b>by </b><i><a class="ref" href="isocat_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x2</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="isocat_1.html#T1" target="_self">Th1</a></i>;<br/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">x1</font></span><a href="cat_2.html#K13">]</a></span> = 
<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_2.html#K18">pr2</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="cat_2.html#K14">.</a> <font color="Maroon">x2</font>,<font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K17">pr1</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="cat_2.html#K14">.</a> <font color="Maroon">x2</font>,<font color="Maroon">x1</font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_3.html#T69">FUNCT_3:69</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cat_2.html#K17">pr1</a> <font color="Maroon">A</font>,<font color="Maroon">B</font></span>)</span> <a href="cat_2.html#K14">.</a> <font color="Maroon">x2</font>,<font color="Maroon">x1</font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="ref" href="funct_3.html#D6">FUNCT_3:def 6</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K13">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="cat_2.html#K13">]</a></span>
<b>by </b><i><a class="ref" href="funct_3.html#D5">FUNCT_3:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E12:22"/>
<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="txt" href="isocat_1.html#E1:22"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>


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