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

<a NAME="E1:9"/><i><font color="Green" title="E5">A1</font></i>: 
2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></i>;<br/>
<a NAME="E2:9"/>
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> 1 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#K8" title="DOMAIN_1:func.8">{</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,1</span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> )
 
<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" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:9_1"/><i><font color="Green" title="E6">A2</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> 1 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#K8" title="DOMAIN_1:func.8">{</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,1</span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> )
 
<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"/><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> 1
 ;<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#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:9_1_1"/><i><font color="Green" title="E8">A4</font></i>: 
( 0 <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> 2 &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> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:9"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="urysohn1.html#E1:9_1_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="urysohn1.html#D3" target="_self" title="URYSOHN1:def.3">Def3</a></i>;<br/>
<a NAME="E5:9_1_1"/>
( 0 <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 or <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 
<b>by </b><i><a class="txt" href="urysohn1.html#E4:9_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/>
<a NAME="E6:9_1_1"/><b>then </b>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T9" title="NAT_1:th.9">NAT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E7:9_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#K8" title="DOMAIN_1:func.8">{</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,1</span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 <b>by </b><i><a class="txt" href="urysohn1.html#E4:9_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:9_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#K8" title="DOMAIN_1:func.8">{</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,1</span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</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> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1_2"/><i><font color="Green" title="E7">A5</font></i>: 
<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#K8" title="DOMAIN_1:func.8">{</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,1</span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_2_1"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2 or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="urysohn1.html#E1:9_1_2"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2_1_1"><b>suppose </b></a><a NAME="E1:9_1_2_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:9_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 ;<br/><b>hence </b><a NAME="E3:9_1_2_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> 1
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:9"><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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2_1_2"><b>suppose </b></a><a NAME="E1:9_1_2_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_1_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> 1
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:9"><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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_2_1_3"><b>suppose </b></a><a NAME="E1:9_1_2_1_3"/><i><font color="Green" title="E8">A6</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><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/><a NAME="E3:9_1_2_1_3"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:9_1_2_1_3"><i><font color="Green" title="E8">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E4:9_1_2_1_3"/>
<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> 1
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:9"><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/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:9_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> 1 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#K8" title="DOMAIN_1:func.8">{</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,1</span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:9_1"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:9"/>
 <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>,1</span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div>
