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

<b>assume </b><a NAME="E1:9"/><i><font color="Green">E24</font></i>: 
not  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K3">RAT</a>  <a href="tarski.html#R2">are_equipotent</a> 
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,  <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K3" target="_self">RAT_with_denominator</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<a NAME="E2:9"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">X</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:9_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K3" target="_self">RAT_with_denominator</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 
<b>by </b><i><a class="ref" href="numbers.html#T12">NUMBERS:12</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">X</font> =  <a href="mesfunc1.html#K3" target="_self">RAT_with_denominator</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a>  <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">X</font>
;<br/>

<b>thus </b><a NAME="E3:9_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">X</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E4:9"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E5:9"/><i><font color="Green">E34</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K3">RAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<a NAME="E1:9_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_2_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:9_2_1"/>
 <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>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:9_2_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</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">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E8:9_2_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K3" target="_self">RAT_with_denominator</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:9_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:9_2"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K3">RAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E3:9_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a>  holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K3">RAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="rat_1.html#NM1">Rational</a> <b>by </b><i><a class="ref" href="rat_1.html#D2">RAT_1:def 2</a></i>;<br/>
<a NAME="E3:9_2_2"/><i><font color="Green">E40</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>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:9_2_2"/>
 <a href="rat_1.html#K1">denominator</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="rat_1.html#D3">RAT_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:9_2_2"/><i><font color="Green">E51</font></i>: 
 <a href="rat_1.html#K1">denominator</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E8:9_2_2"/><i><font color="Green">E52</font></i>: 
 <a href="rat_1.html#K1">denominator</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <span class="p1">(<span class="default"><a href="rat_1.html#K1">denominator</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E10:9_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rat_1.html#K2">numerator</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="rat_1.html#T37">RAT_1:37</a></i>;<br/>
<a NAME="E11:9_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K3" target="_self">RAT_with_denominator</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:9_2_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E13:9_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:9_2"/><b>then </b>
 <a href="numbers.html#K3">RAT</a>  <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:9_2"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K3">RAT</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:9"/>
 <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>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:9"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> is <a href="card_4.html#V1">countable</a>
 
<b>by </b><i><a class="ref" href="card_4.html#T45">CARD_4:45</a></i>;<br/>
<a NAME="E8:9"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> holds <br/><font color="Olive">Y</font> is <a href="card_4.html#V1">countable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_3"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</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">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <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/>
<a NAME="E5:9_3"/>
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K3" target="_self">RAT_with_denominator</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:9_3"/><b>then </b>
 <a href="mesfunc1.html#K2" target="_self">INT</a> ,<font color="Maroon">Y</font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E7:9_3"/><b>then </b>
 <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">Y</font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i>, <a class="ref" href="wellord2.html#T22">WELLORD2:22</a></i>;<br/>
<a NAME="E8:9_3"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="card_1.html#K1">Card</a> <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>
<a NAME="E9:9_3"/>
 <a href="card_1.html#K1">Card</a> <a href="numbers.html#K5">NAT</a>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0
 
<b>by </b><i><a class="ref" href="card_4.html#T44">CARD_4:44</a>, <a class="ref" href="card_4.html#D1">CARD_4:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:9_3"/>
<font color="Maroon">Y</font> is <a href="card_4.html#V1">countable</a>
 <b>by </b><i>, <a class="ref" href="card_4.html#D1">CARD_4:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:9"/><b>then </b>
 <a href="numbers.html#K3">RAT</a>  is <a href="card_4.html#V1">countable</a>
 
<b>by </b><i>, , <a class="ref" href="card_4.html#T62">CARD_4:62</a></i>;<br/>
<a NAME="E10:9"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <a href="numbers.html#K3">RAT</a>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0
 
<b>by </b><i><a class="ref" href="card_4.html#D1">CARD_4:def 1</a></i>;<br/>
<a NAME="E11:9"/><b>then </b>
not  <a href="card_1.html#K3">alef</a> 0 <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <a href="numbers.html#K3">RAT</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T14">CARD_1:14</a></i>;<br/>
<a NAME="E12:9"/><b>then </b>
(  not  <a href="card_1.html#K3">alef</a> 0 <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <a href="numbers.html#K3">RAT</a>  or  not  <a href="card_1.html#K3">alef</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K1">Card</a> <a href="numbers.html#K3">RAT</a>  )
 
<b>by </b><i><a class="ref" href="card_1.html#T13">CARD_1:13</a></i>;<br/>
<a NAME="E13:9"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="card_1.html#K1">Card</a> <a href="numbers.html#K3">RAT</a>  <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K3">alef</a> 0
 
<b>by </b><i>, <a class="ref" href="card_1.html#T14">CARD_1:14</a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a>, <a class="ref" href="card_1.html#T38">CARD_1:38</a></i>;<br/>
<a NAME="E14:9"/>
not  <a href="numbers.html#K3">RAT</a>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="card_1.html#T38">CARD_1:38</a>, <a class="ref" href="card_4.html#T14">CARD_4:14</a>, <a class="ref" href="numbers.html#T18">NUMBERS:18</a></i>;<br/>
<b>hence </b><a NAME="E15:9"/>
contradiction
 <b>by </b><i>, <a class="ref" href="card_4.html#T2">CARD_4:2</a></i>;<br/>


</div>
