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

<b>let </b><font color="Maroon" title="c1">n</font> be    <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> ;<br/>

<a NAME="E1:13"/><i><font color="Green" title="E7">A1</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" title="URYSOHN1:func.5">dyad</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; ( for <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 <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" title="URYSOHN1:func.5">dyad</a> <font color="Maroon" title="c1">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="urysohn1.html#K5" title="URYSOHN1:func.5">dyad</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="urysohn1.html#D6" target="_self" title="URYSOHN1:def.6">Def6</a></i>;<br/>
<a NAME="E2:13"/>
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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" title="URYSOHN1:func.5">dyad</a> <font color="Maroon" title="c1">n</font></span>)</span> iff <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> <font color="Maroon" title="c1">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<a NAME="E1:13_1"/><i><font color="Green" title="E8">A2</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" title="URYSOHN1:func.5">dyad</a> <font color="Maroon" title="c1">n</font></span>)</span> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c1">n</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" title="URYSOHN1:func.5">dyad</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:13_1_1"/><i><font color="Green" title="E9">A3</font></i>: 
( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" title="URYSOHN1:func.5">dyad</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" title="URYSOHN1:func.5">dyad</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:13_1_1"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn1.html#E3:13_1_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="urysohn1.html#D6" target="_self" title="URYSOHN1:def.6">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">y</font> = <font color="Maroon" title="c3">y</font> as    <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>by </b><i><a class="txt" href="urysohn1.html#E3:13_1_1"><i><font color="Green" title="E9">A3</font></i></a></i>;<br/>
<a NAME="E6:13_1_1"/><i><font color="Green" title="E11">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> &amp; <font color="Maroon" title="c4">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="urysohn1.html#E4:13_1_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:13_1_1"/><b>then </b><i><font color="Green" title="E12">A6</font></i>: 
( 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">y</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; <font color="Maroon" title="c4">y</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 )
 
<b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E9:13_1_1"/><i><font color="Green" title="E13">A7</font></i>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E6:13_1_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<a NAME="E10:13_1_1"/><i><font color="Green" title="E14">A8</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E11:13_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn1.html#E3:13_1_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="urysohn1.html#D6" target="_self" title="URYSOHN1:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E12:13_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E7:13_1_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="urysohn1.html#E9:13_1_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="urysohn1.html#E10:13_1_1"><i><font color="Green" title="E14">A8</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>
<a NAME="E2:13_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" title="URYSOHN1:func.5">dyad</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_2"/><i><font color="Green" title="E9">A9</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">x</font> = <font color="Maroon" title="c2">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>consider </b><font color="Maroon" title="c4">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="E4:13_1_2"/><i><font color="Green" title="E10">A10</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="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:13_1_2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="ref" href="urysohn1.html#D3" target="_self" title="URYSOHN1:def.3">Def3</a></i>;<br/>
<a NAME="E5:13_1_2"/><i><font color="Green" title="E11">A11</font></i>: 
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="urysohn1.html#E4:13_1_2"><i><font color="Green" title="E10">A10</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">y</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="E7:13_1_2"/><i><font color="Green" title="E12">A12</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/>
<a NAME="E8:13_1_2"/><i><font color="Green" title="E13">A13</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn1.html#E5:13_1_2"><i><font color="Green" title="E11">A11</font></i></a>, <a class="txt" href="urysohn1.html#E7:13_1_2"><i><font color="Green" title="E12">A12</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:13_1_2"/><b>then </b><i><font color="Green" title="E14">A14</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" title="URYSOHN1:func.5">dyad</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="urysohn1.html#D6" target="_self" title="URYSOHN1:def.6">Def6</a></i>;<br/>
<a NAME="E10:13_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn1.html#E4:13_1_2"><i><font color="Green" title="E10">A10</font></i></a>, <a class="txt" href="urysohn1.html#E7:13_1_2"><i><font color="Green" title="E12">A12</font></i></a></i>;<br/>
<a NAME="E11:13_1_2"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" title="URYSOHN1:func.5">dyad</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>
 
<b>by </b><i><a class="txt" href="urysohn1.html#E1:13"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="urysohn1.html#E8:13_1_2"><i><font color="Green" title="E13">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:13_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" title="URYSOHN1:func.5">dyad</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn1.html#E9:13_1_2"><i><font color="Green" title="E14">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:13_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" title="URYSOHN1:func.5">dyad</a> <font color="Maroon" title="c1">n</font></span>)</span> iff <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:13_1"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:13"/>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" title="URYSOHN1:func.5">dyad</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" title="URYSOHN1:func.5">dyad</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="urysohn1.html#D6" target="_self" title="URYSOHN1:def.6">Def6</a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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