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

<b>thus </b><a NAME="E1:70_1_1"/>
(  not <font color="Maroon">IT</font> is <a href="card_4.html#V1">denumerable</a> or <font color="Maroon">IT</font> is <a href="xboole_0.html#V1">empty</a> or  ex <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">D</font> st <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">F</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:70_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">IT</font> is <a href="card_4.html#V1">denumerable</a>
 <b>and </b><br/><a NAME="E2:70_1_1_1"/><i><font color="Green">E28</font></i>: 
not <font color="Maroon">IT</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:70_1_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E5:70_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="card_4.html#T45">CARD_4:45</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E7:70_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font>
 <b>by </b><i>, <a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<b>set </b><font color="Maroon">F</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span>;<br/>
<a NAME="E8:70_1_1_1"/><i><font color="Green">E50</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:70_1_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/>
<a NAME="E2:70_1_1_1_1"/><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_1_1_1_1"/>
(  <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font> or  <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:70_1_1_1_1_1_1"/><i><font color="Green">E53</font></i>: 
 <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font>
 ;<br/><div class="add"><a NAME="E2:70_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:70_1_1_1_1_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#T16">FUNCOP_1:16</a></i>;<br/><a NAME="E4:70_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E5:70_1_1_1_1_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E6:70_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T32">FUNCT_4:32</a></i>;<br/><b>hence </b> <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T26">RELAT_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="supinf_2.html#E21"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="funcop_1.html#T16">FUNCOP_1:16</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>
<br/>.= 
<font color="Maroon">IT</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>
;<br/><br/><b>thus </b> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="supinf_2.html#E21"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="funcop_1.html#T16">FUNCOP_1:16</a></i>
<br/>.= 
 <a href="numbers.html#K5">NAT</a> 
<b>by </b><i>, <a class="txt" href="supinf_2.html#E20"><i><font color="Green">Lemma61</font></i></a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:70_1_1_1_1_1_2"/><i><font color="Green">E55</font></i>: 
 <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font>
 ;<br/><div class="add"><div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:70_1_1_1_1_1_2_1"/>
<a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><a NAME="E2:70_1_1_1_1_1_2_1"/><b>then </b>
 <a href="numbers.html#K5">NAT</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>hence </b><a NAME="E3:70_1_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="supinf_2.html#E20"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div><a NAME="E3:70_1_1_1_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E4:70_1_1_1_1_1_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T89">XBOOLE_1:89</a></i>;<br/><a NAME="E5:70_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T32">FUNCT_4:32</a></i>;<br/><b>hence </b> <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T26">RELAT_1:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</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="txt" href="supinf_2.html#E21"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</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="relat_1.html#T148">RELAT_1:148</a></i>
<br/>.= 
<font color="Maroon">IT</font> <a href="xboole_0.html#K2">\/</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="txt" href="supinf_2.html#E2:70_1_1_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>
<br/>.= 
<font color="Maroon">IT</font>
<b>by </b><i><a class="txt" href="supinf_2.html#E2:70_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/><br/><b>thus </b> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K8">/\</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
<br/>.= 
 <a href="numbers.html#K5">NAT</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T51">XBOOLE_1:51</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <font color="Maroon">IT</font></span>)</span></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">x</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">D</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>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E10:70_1_1_1"/>
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E2:70_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">IT</font> is <a href="xboole_0.html#V1">empty</a> or  ex <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">D</font> st <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">F</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1_1_2"/>
( <font color="Maroon">IT</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">IT</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_2_1"><b>suppose </b></a><a NAME="E1:70_1_1_2_1"/>
<font color="Maroon">IT</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:70_1_1_2_1"/><b>then </b>
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>hence </b><a NAME="E3:70_1_1_2_1"/>
<font color="Maroon">IT</font> is <a href="card_4.html#V1">denumerable</a>
 <b>by </b><i><a class="ref" href="card_4.html#T43">CARD_4:43</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_2_2"><b>suppose </b></a><a NAME="E1:70_1_1_2_2"/>
not <font color="Maroon">IT</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">D</font><b> such that </b><br/><a NAME="E3:70_1_1_2_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><a NAME="E4:70_1_1_2_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E5:70_1_1_2_2"/>
<font color="Maroon">IT</font> is <a href="card_4.html#V1">denumerable</a>
 <b>by </b><i>, <a class="ref" href="card_4.html#T45">CARD_4:45</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
