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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:14"/><i><font color="Green">E7</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" target="_self">dyad</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" target="_self">dyad</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="urysohn1.html#K5" target="_self">dyad</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="urysohn1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E2:14"/>
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="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" target="_self">dyad</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:14_1"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" target="_self">dyad</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> implies <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" target="_self">dyad</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:14_1_1"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" target="_self">dyad</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" target="_self">dyad</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:14_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn1.html#E3:14_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="urysohn1.html#D6" target="_self">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="urysohn1.html#E3:14_1_1"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E6:14_1_1"/><i><font color="Green">E11</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="urysohn1.html#E4:14_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:14_1_1"/><b>then </b><i><font color="Green">E12</font></i>: 
( 1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 )
 
<b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:14_1_1"/><i><font color="Green">E13</font></i>: 
1 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E6:14_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E10:14_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn1.html#E3:14_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="urysohn1.html#D6" target="_self">Def6</a></i>;<br/>
<b>hence </b><a NAME="E11:14_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E7:14_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="urysohn1.html#E9:14_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="urysohn1.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:14_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" target="_self">dyad</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:14_1_2"/><i><font color="Green">E10</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:14_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="urysohn1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E5:14_1_2"/><i><font color="Green">E11</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="urysohn1.html#E4:14_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:14_1_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>
<a NAME="E8:14_1_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn1.html#E5:14_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="urysohn1.html#E7:14_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:14_1_2"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" target="_self">dyad</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="urysohn1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E10:14_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn1.html#E4:14_1_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="urysohn1.html#E7:14_1_2"><i><font color="Green">E12</font></i></a></i>;<br/>
<a NAME="E11:14_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" target="_self">dyad</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="urysohn1.html#E1:14"><i><font color="Green">E7</font></i></a>, <a class="txt" href="urysohn1.html#E8:14_1_2"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:14_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" target="_self">dyad</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn1.html#E9:14_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:14_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" target="_self">dyad</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> iff <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:14_1"><i><font color="Green">E8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:14"/>
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" target="_self">dyad</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K5" target="_self">dyad</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="urysohn1.html#D6" target="_self">Def6</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
