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

<a NAME="E1:8"/><i><font color="Green">E23</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="E2:8"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0 iff  ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_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/>

<a NAME="E1:8_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0 implies  ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1_1"/>
<font color="Maroon">x</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">i</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="E3:8_1_1"/><i><font color="Green">E32</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="real_1.html#K6">/</a> 1 )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E4:8_1_1"/>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:8_1"/>
(  ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> ) implies <font color="Maroon">x</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="8_1_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:8_1_2"/><i><font color="Green">E39</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> )
 ;<br/>

<a NAME="E2:8_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="real_1.html#K6">/</a> 1
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:8_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:8_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0 iff  ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:8"/>
 <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="8_2"><b>proof </b></a><div class="add">

<a NAME="E1:8_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0 iff <font color="Olive">x</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="8_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/>

<a NAME="E1:8_2_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0 implies <font color="Maroon">x</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="8_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_2_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</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">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>consider </b><font color="Maroon">i</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="E4:8_2_1_1"/><i><font color="Green">E42</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:8_2_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i/>;<br/>
<a NAME="E6:8_2_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/>
<b>hence </b><a NAME="E7:8_2_1_1"/>
<font color="Maroon">x</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="E2:8_2_1"/>
( <font color="Maroon">x</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">x</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="8_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_2_1_2"/>
<font color="Maroon">x</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="E2:8_2_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</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="E3:8_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:8_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> 0 iff <font color="Maroon">x</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/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:8_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="E4:8"/>
 <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>
