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

<b>let </b><font color="Maroon">s1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">s2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:10_1_2"/><i><font color="Green">E42</font></i>: 
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> 0 implies <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> 0 ) &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Olive">n</font> holds <br/><font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>and </b><br/><a NAME="E2:10_1_2"/><i><font color="Green">E44</font></i>: 
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> 0 implies <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> 0 ) &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Olive">n</font> holds <br/><font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 ) )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1_2_1"/>
( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_1"><b>suppose </b></a><a NAME="E1:10_1_2_1_1"/>
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_2_1_1"/>
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_2"><b>suppose </b></a><a NAME="E1:10_1_2_1_2"/><i><font color="Green">E45</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font>
 ;<br/><div class="add"><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> ] <b>means </b><font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">a<sub>1</sub></font>;<br/><a NAME="E2:10_1_2_1_2"/><i><font color="Green">E46</font></i>: 
 ex <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>]
 <b>by </b><i>, <a class="ref" href="urysohn1.html#D4">URYSOHN1:def 4</a></i>;<br/><a NAME="E3:10_1_2_1_2"/>
 ex <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>] &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i><b>then consider </b><font color="Maroon">q</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="E5:10_1_2_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">q</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Olive">i</font> holds <br/><font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> ) )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_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="E1:10_1_2_1_2_1_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:10_1_2_1_2_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_2_1_2_1_1_1"/>
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:10_1_2_1_2_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font>
 ;<br/><div class="add"><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="E3:10_1_2_1_2_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">q</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="E5:10_1_2_1_2_1_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">m</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_2_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1_2_1_2_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E2:10_1_2_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:10_1_2_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:10_1_2_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E7:10_1_2_1_2_1_1_2"/>
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:10_1_2_1_2"/>
<font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
