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

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

<b>thus </b><a NAME="E1:67"/>
( <font color="Maroon">A</font> is <a href="nattra_1.html#V2" target="_self">discrete</a> implies ( ( for <font color="Olive">a</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>  ex <font color="Olive">B</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">a</font>,<font color="Olive">a</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> 1 ) ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>  st <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> holds <br/> <a href="cat_1.html#K6">Hom</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">A</font> is <a href="nattra_1.html#V2" target="_self">discrete</a>
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/><a NAME="E1:67_1_1"/><i><font color="Green">E8</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="cat_1.html#K10">id</a> <font color="Maroon">a</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">B</font> =  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">B</font> = <font color="Maroon">B</font>;<br/><b>thus </b><a NAME="E3:67_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">a</font>
 ;<br/><b>thus </b><a NAME="E4:67_1_1"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">a</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> be   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E3:67_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/>

<b>assume </b><a NAME="E4:67_1"/><i><font color="Green">E26</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E8:67_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <font color="Maroon">c</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:67_1"/>
 <a href="cat_1.html#K10">id</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="cat_1.html#T55">CAT_1:55</a></i>;<br/>
<a NAME="E10:67_1"/><b>then </b>
 <a href="cat_1.html#K6">Hom</a> <font color="Maroon">c</font>,<font color="Maroon">c</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="cat_1.html#K6">Hom</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E11:67_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:67_1"/>
contradiction
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E2:67"/><i><font color="Green">E33</font></i>: 
( ( for <font color="Olive">a</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>  ex <font color="Olive">B</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <font color="Olive">a</font>,<font color="Olive">a</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> 1 ) ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="cat_1.html#NM1">Object</a> of <font color="Maroon">A</font>  st <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> holds <br/> <a href="cat_1.html#K6">Hom</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="cat_1.html#NM2">Morphism</a> of <font color="Maroon">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="nattra_1.html#D19" target="_self">NATTRA_1:def 19</a><br/>

<b>take </b>
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font>
;<br/>

<b>consider </b><font color="Maroon">B</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:67"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i/>;<br/>
<a NAME="E5:67"/>
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="cat_1.html#T19">CAT_1:19</a></i>;<br/>
<a NAME="E6:67"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:67"/><i><font color="Green">E36</font></i>: 
 <a href="cat_1.html#K6">Hom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/>
<a NAME="E9:67"/>
( <font color="Maroon">f</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> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> <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> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="cat_1.html#T55">CAT_1:55</a></i>;<br/>
<a NAME="E10:67"/><b>then </b>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:67"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K10">id</a> <span class="p1">(<span class="default"><a href="cat_1.html#K2">dom</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>


</div>
