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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="cat_1.html#NM4">Category</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  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>;<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#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> ;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = 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>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<div><i><font color="Green">E20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E4:32_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>10</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#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="cat_2.html#E1:32"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E5:32_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="cat_2.html#E4:32_1"><i><font color="Green">E22</font></i></a></i>;<br/><b>hence </b><a NAME="E6:32_1"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="cat_2.html#E3:32_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="cat_1.html#T18">CAT_1:18</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:32"/><i><font color="Green">E21</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 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="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:32"/><i><font color="Green">E22</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E5:32"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is  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="txt" href="cat_2.html#E1:32"><i><font color="Green">E19</font></i></a>, <a class="ref" href="cat_2.html#T28" target="_self">Th28</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:32_2"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="cat_2.html#E3:32"><i><font color="Green">E21</font></i></a>, <a class="txt" href="cat_2.html#E5:32"><i><font color="Green">E23</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><b>thus </b><a NAME="E2:32_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</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="E1:32_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:32_2_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:32_2_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="cat_2.html#E3:32"><i><font color="Green">E21</font></i></a>, <a class="txt" href="cat_2.html#E4:32"><i><font color="Green">E22</font></i></a>, <a class="txt" href="cat_2.html#E3:32_2_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:32_2_1"/>
( <span class="p1">(<span class="default">the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E4:32"><i><font color="Green">E22</font></i></a>, <a class="txt" href="cat_2.html#E3:32_2_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E7:32_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cat_2.html#E2:32"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cat_2.html#E4:32_2_1"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:32_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</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="E1:32_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:32_2_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:32_2_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="cat_2.html#E3:32"><i><font color="Green">E21</font></i></a>, <a class="txt" href="cat_2.html#E4:32"><i><font color="Green">E22</font></i></a>, <a class="txt" href="cat_2.html#E3:32_2_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:32_2_2"/>
( <span class="p1">(<span class="default">the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E4:32"><i><font color="Green">E22</font></i></a>, <a class="txt" href="cat_2.html#E3:32_2_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E7:32_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cat_2.html#E2:32"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cat_2.html#E4:32_2_2"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:32"/>
( the <a href="cat_1.html#U3">Dom</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; the <a href="cat_1.html#U4">Cod</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></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/>

<b>thus </b><a NAME="E8:32"/>
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>4</sub></font> is   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>proof </b></a><div class="add">

<a NAME="E1:32_3"/><i><font color="Green">E24</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</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">c<sub>1</sub></font>,the <a href="cat_1.html#U2">Morphisms</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E2:32_3"/><i><font color="Green">E25</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">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>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E3:32_3"/><b>then </b><i><font color="Green">E26</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">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>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E4:32_3"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">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>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</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="E1:32_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">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>4</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:32_3_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">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>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:32_3_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">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>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:32_3_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="cat_2.html#E2:32_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="cat_2.html#E3:32_3_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:32_3_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="cat_2.html#E1:32_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1"><b>now </b></a><div class="add"><b>thus </b><span class="p1">(<span class="default">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>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> = 
the <a href="cat_1.html#U5">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="cat_2.html#E3:32_3_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="cat_2.html#E7:32_3_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="cat_2.html#E5:32_3_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="cat_2.html#E7:32_3_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="cat_1.html#D4">CAT_1:def 4</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3_1_1_2"><b>now </b></a><div class="add"><a NAME="E1:32_3_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="cat_2.html#E2:32_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="cat_2.html#E3:32_3_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="cat_2.html#E7:32_3_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E2:32_3_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E3:32_3_1_1_2"/>
(  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="cat_2.html#E2:32"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E4:32_3_1_1_2"/>
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="cat_2.html#E5:32_3_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="cat_2.html#E7:32_3_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="cat_1.html#T40">CAT_1:40</a></i>;<br/><b>hence </b><a NAME="E5:32_3_1_1_2"/>
(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="cat_1.html#T42">CAT_1:42</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:32_3_1_1"/>
(  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="cat_1.html#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  &amp; <font color="Maroon">c<sub>11</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="cat_1.html#K4">*</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:32_3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cat_2.html#E1:32"><i><font color="Green">E19</font></i></a>, <a class="txt" href="cat_2.html#E4:32_3_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:32_3"/>
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>4</sub></font> is   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cat_2.html#E3:32_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>


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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4"><b>now </b></a><div class="add"><a NAME="E1:32_4"/><i><font color="Green">E24</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 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="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E2:32_4"/><i><font color="Green">E25</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/><b>thus </b><a NAME="E3:32_4"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cat_2.html#E1:32_4"><i><font color="Green">E24</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><b>thus </b><a NAME="E4:32_4"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</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="E1:32_4_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:32_4_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:32_4_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="cat_2.html#E2:32_4"><i><font color="Green">E25</font></i></a>, <a class="txt" href="cat_2.html#E3:32_4_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E6:32_4_1"/>
( <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="cat_2.html#E2:32_4"><i><font color="Green">E25</font></i></a>, <a class="txt" href="cat_2.html#E3:32_4_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E7:32_4_1"/><b>then </b>
( <span class="p1">(<span class="default">the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp;  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>11</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#K6">Hom</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="cat_1.html#T55">CAT_1:55</a></i>;<br/>
<b>hence </b><a NAME="E8:32_4_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cat_2.html#E1:32"><i><font color="Green">E19</font></i></a>, <a class="txt" href="cat_2.html#E4:32_4_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:32"/>
the <a href="cat_1.html#U6">Id</a> of <font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>2</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></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>
