<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>now </b></a><div class="add"><a NAME="E1:78_1"/>
( the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="nattra_1.html#D20" target="_self">Def20</a></i>;<br/><b>hence </b><i><font color="Green">E58</font></i>: <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U1">Objects</a> of <span class="p2">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="cat_1.html#U1">Objects</a> of <span class="p2">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> = 
the <a href="cat_1.html#U1">Objects</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span>

<br/>.= 
the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="nattra_1.html#D20" target="_self">Def20</a></i>
;<br/><br/><b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span> : verum </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font> : verum </span>}</span> ;<br/><a NAME="E3:78_1"/><i><font color="Green">E59</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span> : verum </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> ,<span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font> : verum </span>}</span> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:78_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> ,<span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font> : verum </span>}</span> </span><a href="zfmisc_1.html#K2">:]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></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="E1:78_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span><b> such that </b><br/><a NAME="E3:78_1_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:78_1_2_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="cat_2.html#K12">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E6:78_1_2_1"/><i><font color="Green">E62</font></i>: 
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</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> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E5:78_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>;<br/>
<a NAME="E7:78_1_2_1"/>
(  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  &amp;  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font> : verum </span>}</span>  )
 
;<br/>
<b>hence </b><a NAME="E8:78_1_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> ,<span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font> : verum </span>}</span> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E3:78_1_2_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="nattra_1.html#E6:78_1_2_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>6</sub></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="E2:78_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> ,<span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font> : verum </span>}</span> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:78_1_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font> : verum </span>}</span>  )
 <b>and </b><br/><a NAME="E5:78_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103">ZFMISC_1:103</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:78_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E4:78_1_2"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:78_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="nattra_1.html#E4:78_1_2"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E10:78_1_2"/>
 <a href="cat_1.html#K10">id</a> <span class="p1"><a href="cat_2.html#K12">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="cat_2.html#K12">]</a></span> <a href="hidden.html#R1">=</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> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="cat_2.html#K13">]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#T96">FUNCT_3:96</a></i>;<br/>
<b>hence </b><a NAME="E11:78_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="nattra_1.html#E5:78_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="nattra_1.html#E7:78_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="nattra_1.html#E9:78_1_2"><i><font color="Green">E63</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:78_1"/>
( the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  &amp; the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>2</sub></font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="nattra_1.html#D20" target="_self">Def20</a></i>;<br/><b>hence </b><a NAME="E5:78_1"/><i><font color="Green">E60</font></i>: 
the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="cat_1.html#U2">Morphisms</a> of <span class="p2">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="cat_1.html#U2">Morphisms</a> of <span class="p2">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E3:78_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="nattra_1.html#D20" target="_self">Def20</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="cat_2.html#T13">CAT_2:13</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="cat_2.html#T13">CAT_2:13</a></i>;<br/><a NAME="E8:78_1"/>
( the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K7">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K7">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="nattra_1.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><span class="p1"><a href="cat_2.html#K9">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <span class="p2">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="cat_1.html#U3">Dom</a> of <span class="p2">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_2.html#K9">:]</a></span> = 
<span class="p1"><a href="cat_2.html#K9">[:</a><span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K9">:]</a></span> <a href="nattra_1.html#K1" target="_self">|</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
<b>by </b><i><a class="ref" href="nattra_1.html#T1" target="_self">Th1</a></i>
<br/>.= 
the <a href="cat_1.html#U3">Dom</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span> <a href="relat_1.html#K7">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:78_1"><i><font color="Green">E60</font></i></a></i>
<br/>.= 
the <a href="cat_1.html#U3">Dom</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="nattra_1.html#T8" target="_self">Th8</a></i>
;<br/><br/><a NAME="E10:78_1"/>
( the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K7">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K7">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="nattra_1.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><span class="p1"><a href="cat_2.html#K9">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <span class="p2">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="cat_1.html#U4">Cod</a> of <span class="p2">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_2.html#K9">:]</a></span> = 
<span class="p1"><a href="cat_2.html#K9">[:</a><span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K9">:]</a></span> <a href="nattra_1.html#K1" target="_self">|</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
<b>by </b><i><a class="ref" href="nattra_1.html#T1" target="_self">Th1</a></i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span> <a href="relat_1.html#K7">|</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:78_1"><i><font color="Green">E60</font></i></a></i>
<br/>.= 
the <a href="cat_1.html#U4">Cod</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="nattra_1.html#T8" target="_self">Th8</a></i>
;<br/><br/><i><font color="Green">E61</font></i>: <span class="p1"><a href="cat_2.html#K10">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K10">:|</a></span> <a href="realset1.html#K1">||</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="mcart_1.html#K12">:]</a></span> = 
the <a href="cat_1.html#U5">Comp</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span> <a href="realset1.html#K1">||</a> the <a href="cat_1.html#U2">Morphisms</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:78_1"><i><font color="Green">E60</font></i></a></i>
<br/>.= 
the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="nattra_1.html#T8" target="_self">Th8</a></i>
;<br/><a NAME="E13:78_1"/>
( the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>6</sub></font> &amp; the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>2</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="nattra_1.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><a NAME="E14:78_1"/>
<span class="p1"><a href="cat_2.html#K10">|:</a><span class="default">the <a href="cat_1.html#U5">Comp</a> of <span class="p2">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="cat_1.html#U5">Comp</a> of <span class="p2">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_2.html#K10">:|</a></span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E12:78_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nattra_1.html#T2" target="_self">Th2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="cat_2.html#D4">CAT_2:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="cat_1.html#U1">Objects</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="cat_2.html#D4">CAT_2:def 4</a></i>;<br/><a NAME="E17:78_1"/>
( the <a href="cat_1.html#U6">Id</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_1.html#K7">|</a> the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; the <a href="cat_1.html#U6">Id</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>2</sub></font> <a href="relat_1.html#K7">|</a> the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="nattra_1.html#T8" target="_self">Th8</a></i>;<br/><b>hence </b><span class="p1"><a href="cat_2.html#K9">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <span class="p2">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,the <a href="cat_1.html#U6">Id</a> of <span class="p2">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_2.html#K9">:]</a></span> = 
<span class="p1"><a href="cat_2.html#K9">[:</a><span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K9">:]</a></span> <a href="nattra_1.html#K1" target="_self">|</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
<b>by </b><i><a class="ref" href="nattra_1.html#T1" target="_self">Th1</a></i>
<br/>.= 
the <a href="cat_1.html#U6">Id</a> of <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span> <a href="relat_1.html#K7">|</a> the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="nattra_1.html#E2:78_1"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
the <a href="cat_1.html#U6">Id</a> of <span class="p1">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <span class="p2"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="nattra_1.html#T8" target="_self">Th8</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:78"/>
 <a href="nattra_1.html#K13" target="_self">IdCat</a> <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="cat_2.html#K11">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K11">[:</a><span class="default"><span class="p2">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="nattra_1.html#K13" target="_self">IdCat</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="cat_2.html#K11">:]</a></span>
 ;<br/>


</div>
