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

<a NAME="E1:8"/><i><font color="Green" title="E5">A1</font></i>: 
2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/>
<a NAME="E2:8"/><i><font color="Green" title="E6">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">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" title="c1">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<a NAME="E1:8_1"/><i><font color="Green" title="E7">A3</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:8_1_1"/><i><font color="Green" title="E8">A4</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 1 )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:8"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="urysohn1.html#D3" target="_self" title="URYSOHN1:def.3">Def3</a></i>;<br/>
<b>thus </b><a NAME="E4:8_1_1"/>
 ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E3:8_1_1"><i><font color="Green" title="E8">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:8_1"/>
(  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> ) implies <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<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" title="c2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><a NAME="E2:8_1_2"/><i><font color="Green" title="E8">A5</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> )
 ;<br/>

<a NAME="E3:8_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 1
 
<b>by </b><i><a class="txt" href="urysohn1.html#E2:8_1_2"><i><font color="Green" title="E8">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:8_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:8"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="urysohn1.html#E2:8_1_2"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="urysohn1.html#D3" target="_self" title="URYSOHN1:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:8_1"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> ) )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:8_1"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:8"/>
 <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</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" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</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" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:8_2_1"/><i><font color="Green" title="E7">A6</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</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" title="E8">A7</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c2">x</font> = <font color="Maroon" title="c1">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>consider </b><font color="Maroon" title="c3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:8_2_1_1"/><i><font color="Green" title="E9">A8</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E2:8"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="urysohn1.html#E1:8_2_1_1"><i><font color="Green" title="E8">A7</font></i></a></i>;<br/>
<a NAME="E5:8_2_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="urysohn1.html#E4:8_2_1_1"><i><font color="Green" title="E9">A8</font></i></a></i>;<br/>
<a NAME="E6:8_2_1_1"/><b>then </b>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="urysohn1.html#E4:8_2_1_1"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="nat_1.html#T9" title="NAT_1:th.9">NAT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E7:8_2_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:8_2_1"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<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" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ;<br/>

<a NAME="E2:8_2_1_2"/><b>then </b>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:8_2_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="urysohn1.html#E2:8"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:8_2_1"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:8_2_1"><i><font color="Green" title="E7">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:8_2"/>
 <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:8"/>
 <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ;<br/>


</div>
