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

<a NAME="E1:9"/><i><font color="Green">E5</font></i>: 
2 <a href="card_4.html#K1">|^</a> 0 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/>
<a NAME="E2:9"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0 iff ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<a NAME="E1:9_1"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0 implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:9_1_1"/><i><font color="Green">E8</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K6">/</a> 1 )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:9"><i><font color="Green">E5</font></i></a>, <a class="ref" href="urysohn1.html#D3" target="_self">Def3</a></i>;<br/>
<b>thus </b><a NAME="E4:9_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E3:9_1_1"><i><font color="Green">E8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:9_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><a NAME="E1:9_1_2"/><i><font color="Green">E8</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:9_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K6">/</a> 1
 
<b>by </b><i><a class="txt" href="urysohn1.html#E1:9_1_2"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:9_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:9"><i><font color="Green">E5</font></i></a>, <a class="txt" href="urysohn1.html#E1:9_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="urysohn1.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:9_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0 iff ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:9_1"><i><font color="Green">E7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:9"/>
 <a href="urysohn1.html#K1" target="_self">dyadic</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 
<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">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0 iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> )
 
<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">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:9_2_1"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0 implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_2_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:9_2_1_1"/><i><font color="Green">E9</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E2:9"><i><font color="Green">E6</font></i></a>, <a class="txt" href="urysohn1.html#E1:9_2_1_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E5:9_2_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="urysohn1.html#E4:9_2_1_1"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E6:9_2_1_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="urysohn1.html#E4:9_2_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/>
<b>hence </b><a NAME="E7:9_2_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:9_2_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_2_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<a NAME="E2:9_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:9_2_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0
 <b>by </b><i><a class="txt" href="urysohn1.html#E2:9"><i><font color="Green">E6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:9_2_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0 iff <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:9_2_1"><i><font color="Green">E7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:9_2"/>
 <a href="urysohn1.html#K1" target="_self">dyadic</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:9"/>
 <a href="urysohn1.html#K1" target="_self">dyadic</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default">0,1</span><a href="domain_1.html#K7">}</a></span>
 ;<br/>


</div>
