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

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:23"/><i><font color="Green">E20</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font> )
 ;<br/>

<b>assume </b><a NAME="E2:23"/><i><font color="Green">E23</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 <br/>(  not <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">n</font> or  not <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="real_1.html#NM1">Real</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="real_1.html#NM1">Real</a>] <b>means </b>( <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  );<br/>
<b>consider </b><font color="Maroon">N</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E4:23"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">n</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] )
 <b>from </b><i><a class="ref" href="real_1.html#S1">REAL_1:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon">M</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E6:23"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">n</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] )
 <b>from </b><i><a class="ref" href="real_1.html#S1">REAL_1:sch 1</a>();<br/></i>
<a NAME="E7:23"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">N</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="supinf_1.html#K3">REAL</a> 
 
;<br/>
<a NAME="E8:23"/>
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<a NAME="E1:23_1"/>
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> holds <br/><font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="urysohn2.html#E2:23"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:23_1"/>
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">N</font> = <font color="Maroon">N</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E10:23"/>
<font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </b></a><div class="add">

<a NAME="E1:23_2"/>
for <font color="Olive">n</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> holds <br/><font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="txt" href="urysohn2.html#E4:23"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:23_2"/>
<font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E12:23"/><i><font color="Green">E27</font></i>: 
 <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3"><b>proof </b></a><div class="add">

<a NAME="E1:23_3"/>
 <a href="numbers.html#K5">NAT</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">A</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E2:23_3_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E6:23"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E3:23_3_1"/><i><font color="Green">E29</font></i>: 
( 0 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">x</font> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:23_3_1_1"/><i><font color="Green">E30</font></i>: 
0 <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="txt" href="urysohn2.html#E2:23"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E2:23_3_1_1"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">x</font> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:23_3_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">A</font>
 ;<br/>

<a NAME="E2:23_3_1_1_1"/>
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_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="E1:23_3_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 </span><b>by </b><i><a class="txt" href="urysohn2.html#E12:23"><i><font color="Green">E27</font></i></a>, <a class="txt" href="urysohn2.html#E1:23_3_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:23_3_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_1_1_1_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="E1:23_3_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> or <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:23_3_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:23_3_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E2:23"><i><font color="Green">E23</font></i></a></i>;<br/><b>hence </b><a NAME="E3:23_3_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E12:23"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:23_3_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:23_3_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E4:23"><i><font color="Green">E24</font></i></a></i>;<br/><b>hence </b><a NAME="E3:23_3_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E12:23"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:23_3_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:23_3_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_3_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> or <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_1_1_1_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:23_3_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:23_3_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E2:23"><i><font color="Green">E23</font></i></a></i>;<br/><b>hence </b><a NAME="E3:23_3_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E12:23"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_3_1_1_1_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:23_3_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:23_3_1_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E4:23"><i><font color="Green">E24</font></i></a></i>;<br/><b>hence </b><a NAME="E3:23_3_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E12:23"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:23_3_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:23_3_1_1_1"/>
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:23_3_1_1"/>
( 0 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">x</font> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> ) )
 <b>by </b><i><a class="txt" href="urysohn2.html#E12:23"><i><font color="Green">E27</font></i></a>, <a class="txt" href="urysohn2.html#E3:23_3_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:23_3_1"/>
 <a href="numbers.html#K5">NAT</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:23_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:23_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T2">NAT_1:2</a>, <a class="txt" href="urysohn2.html#E2:23_3_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:23_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:23_3_1"/>
 <a href="numbers.html#K5">NAT</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="txt" href="urysohn2.html#E12:23"><i><font color="Green">E27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:23_3"/>
 <a href="numbers.html#K5">NAT</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">M</font>
 <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="E13:23"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">n</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span>;<br/>
<a NAME="E1:23_4"/><i><font color="Green">E32</font></i>: 
( <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> )
 
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E2:23_4"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_4_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 1 or 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_1_1_1"><b>suppose </b></a><a NAME="E1:23_4_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:23_4_1_1_1"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 0 )
 <b>by </b><i>, <a class="ref" href="real_2.html#T105">REAL_2:105</a></i>;<br/><b>hence </b><a NAME="E3:23_4_1_1_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_1_1_2"><b>suppose </b></a><a NAME="E1:23_4_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><a NAME="E2:23_4_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">a</font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="real_1.html#K5">-</a> 1 )
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_4_1_1_2_1_1"/>
( <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:23_4_1_1_2_1_1_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_4_1_1_2_1_1_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:23_4_1_1_2_1_1_2"/><i><font color="Green">E55</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:23_4_1_1_2_1_1_2"/><i><font color="Green">E58</font></i>: 
( <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="int_1.html#T8">INT_1:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_1_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_4_1_1_2_1_1_2_1_1"/>
( <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">k</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_1_1_2_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:23_4_1_1_2_1_1_2_1_1_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_4_1_1_2_1_1_2_1_1_1"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_4_1_1_2_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:23_4_1_1_2_1_1_2_1_1_2"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:23_4_1_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:23_4_1_1_2_1_1_2_1_1_2"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:23_4_1_1_2_1_1_2"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:23_4_1_1_2"/>
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:23_4"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="txt" href="urysohn2.html#E12:23"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Maroon">a</font></span><a href="int_1.html#K1">/]</a></span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E5:23_4"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 <b>by </b><i><a class="txt" href="urysohn2.html#E2:23"><i><font color="Green">E23</font></i></a>, <a class="txt" href="urysohn2.html#E12:23"><i><font color="Green">E27</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E15:23"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; not <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 ;<br/>
<a NAME="E16:23"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="txt" href="urysohn2.html#E7:23"><i><font color="Green">E26</font></i></a>, <a class="txt" href="urysohn2.html#E12:23"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E17:23"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 
;<br/>
<a NAME="E18:23"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1] )
 
<b>by </b><i><a class="txt" href="urysohn2.html#E2:23"><i><font color="Green">E23</font></i></a>, <a class="txt" href="urysohn2.html#E4:23"><i><font color="Green">E24</font></i></a>, <a class="txt" href="urysohn2.html#E12:23"><i><font color="Green">E27</font></i></a>, </i>;<br/>
<a NAME="E19:23"/><b>then </b>
<font color="Maroon">n</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="real_1.html#K3">+</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="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E20:23"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T23">XREAL_1:23</a></i>;<br/>


</div>
