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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:41"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">T</font> <a href="compts_1.html#NR4" target="_self">is_T4</a> 
 ;<br/>

<b>let </b><font color="Maroon">A</font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">B</font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E2:41"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">G</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font>, <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E3:41"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E4:41"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">r1</font>, <font color="Olive">r2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font>  st <font color="Olive">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r2</font> holds <br/>( <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Olive">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Olive">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Olive">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Olive">r2</font> )
 ;<br/>

<a NAME="E5:41"/><i><font color="Green">E39</font></i>: 
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="urysohn1.html#D3" target="_self">Def3</a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E6:41"/><i><font color="Green">E40</font></i>: 
( 0 <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E7:41"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font>  holds <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Olive">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1"/>
( 0 <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_1"/>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_1_1"/>
<font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_2"/><i><font color="Green">E42</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">q1</font> = 0 as    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/><a NAME="E3:41_1_1_2"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r</font> )
 <b>by </b><i>, , <a class="txt" href="urysohn1.html#E3:41"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><a NAME="E4:41_1_1_2"/><b>then </b>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:41_1_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E6:41_1_1_2"/>
<font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">S</font> = <span class="p1">(<span class="default"><a href="urysohn1.html#K1" target="_self">dyadic</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>,  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>] <b>means </b>for <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> holds <br/>for <font color="Olive">r1</font>, <font color="Olive">r2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font>  st <font color="Olive">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Olive">r</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Olive">r2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Olive">r</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> holds <br/><font color="Maroon">a<sub>2</sub></font> is    <a href="urysohn1.html#M1" target="_self">Between</a> of <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Olive">r1</font>,<font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Olive">r2</font>;<br/>
<a NAME="E9:41"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font>;<br/>

<a NAME="E1:41_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:41_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="urysohn1.html#E4:41"><i><font color="Green">E32</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">q1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font>, <font color="Maroon">q2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E5:41_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:41_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/>
<b>consider </b><font color="Maroon">Q0</font> being    <a href="urysohn1.html#M1" target="_self">Between</a> of <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font>,<font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font>;<br/>
<b>take </b>
<font color="Maroon">Q0</font>
;<br/>

<b>thus </b><a NAME="E8:41_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">Q0</font>]
 <b>by </b><i><a class="ref" href="urysohn1.html#D4" target="_self">Def4</a>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">D</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">S</font>, <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E11:41"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="urysohn1.html#E3:41"><i><font color="Green">E31</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>] <b>means </b>for <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> holds <br/>( ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> ) &amp; ( not <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> ) );<br/>
<a NAME="E12:41"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_1_1"><b>suppose </b></a><a NAME="E1:41_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> ;<br/><b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:41_3_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">x</font>
 ;<br/><b>take </b>
<font color="Maroon">y</font>
;<br/><b>thus </b><a NAME="E5:41_3_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_1_2"><b>suppose </b></a><a NAME="E1:41_3_1_2"/><i><font color="Green">E89</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:41_3_1_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/><b>take </b>
<font color="Maroon">y</font>
;<br/><b>thus </b><a NAME="E5:41_3_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E14:41"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="urysohn1.html#D4" target="_self">Def4</a>);<br/></i>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<a NAME="E15:41"/>
( 0 <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E16:41"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> )
 <b>by </b><i>, <a class="txt" href="urysohn1.html#E2:41"><i><font color="Green">E29</font></i></a>, </i>;<br/>

<b>let </b><font color="Maroon">r1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <font color="Maroon">r2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <font color="Maroon">r</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>

<b>assume </b><a NAME="E17:41"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font>
 ;<br/>

<b>thus </b><a NAME="E18:41"/>
( <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_4_1_1"/>
( ( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> ) or ( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> ) or ( not <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> ) or ( not <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_1_1"><b>suppose </b></a><a NAME="E1:41_4_1_1_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><a NAME="E2:41_4_1_1_1"/><b>then </b><i><font color="Green">E94</font></i>: 
( <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">r1</font>, <font color="Maroon">r2</font> = <font color="Maroon">r2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/><a NAME="E4:41_4_1_1_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:41_4_1_1_1"/>
( <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i>, <a class="ref" href="urysohn1.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_1_2"><b>suppose </b></a><a NAME="E1:41_4_1_1_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><a NAME="E2:41_4_1_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:41_4_1_1_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_4_1_1_2"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">r2</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">r2</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">q1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font>, <font color="Maroon">q2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E6:41_4_1_1_2"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">r2</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">r2</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E7:41_4_1_1_2"/>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> is    <a href="urysohn1.html#M1" target="_self">Between</a> of <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font>,<font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E4:41"><i><font color="Green">E32</font></i></a>, <a class="ref" href="urysohn1.html#T5" target="_self">Th5</a>, <a class="txt" href="urysohn1.html#E5:41"><i><font color="Green">E39</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">D0</font> being    <a href="urysohn1.html#M1" target="_self">Between</a> of <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font>,<font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font><b> such that </b><br/><a NAME="E9:41_4_1_1_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">D0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font>
 ;<br/><a NAME="E10:41_4_1_1_2"/>
<font color="Maroon">q1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E5:41"><i><font color="Green">E39</font></i></a>, </i>;<br/><a NAME="E11:41_4_1_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
( <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E12:41_4_1_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
( <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font> )
 <b>by </b><i>, <a class="ref" href="urysohn1.html#T6" target="_self">Th6</a>, <a class="txt" href="urysohn1.html#E6:41"><i><font color="Green">E40</font></i></a>, <a class="txt" href="urysohn1.html#E1:41"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E13:41_4_1_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font>
 <b>by </b><i>, <a class="txt" href="urysohn1.html#E5:41"><i><font color="Green">E39</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_4_1_1_2_1_1"/>
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> or <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q1</font> )
 </span><b>by </b><i><a class="ref" href="urysohn1.html#T9" target="_self">Th9</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:41_4_1_1_2_1_1_1"/>
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_4_1_1_2_1_1_1"/>
( <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i>, <a class="txt" href="urysohn1.html#E7:41"><i><font color="Green">E41</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:41_4_1_1_2_1_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">q0</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:41_4_1_1_2_1_1_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">q0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font>
 <b>by </b><i/>;<br/><a NAME="E4:41_4_1_1_2_1_1_2"/>
( <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q0</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q0</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="urysohn1.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E5:41_4_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i>, , <a class="ref" href="urysohn1.html#T6" target="_self">Th6</a>, <a class="txt" href="urysohn1.html#E6:41"><i><font color="Green">E40</font></i></a>, <a class="txt" href="urysohn1.html#E7:41"><i><font color="Green">E41</font></i></a>, <a class="ref" href="urysohn1.html#D6" target="_self">Def6</a>, <a class="txt" href="urysohn1.html#E1:41"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><a NAME="E6:41_4_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E7:41_4_1_1_2_1_1_2"/>
( <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E15:41_4_1_1_2"/>
( <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_1_3"><b>suppose </b></a><a NAME="E1:41_4_1_1_3"/><i><font color="Green">E118</font></i>: 
( not <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><a NAME="E2:41_4_1_1_3"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:41_4_1_1_3"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_4_1_1_3"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">r1</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">r1</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">q1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font>, <font color="Maroon">q2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E6:41_4_1_1_3"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">r1</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">r1</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E7:41_4_1_1_3"/>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> is    <a href="urysohn1.html#M1" target="_self">Between</a> of <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font>,<font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E4:41"><i><font color="Green">E32</font></i></a>, <a class="ref" href="urysohn1.html#T5" target="_self">Th5</a>, <a class="txt" href="urysohn1.html#E5:41"><i><font color="Green">E39</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">D0</font> being    <a href="urysohn1.html#M1" target="_self">Between</a> of <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font>,<font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font><b> such that </b><br/><a NAME="E9:41_4_1_1_3"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">D0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font>
 ;<br/><a NAME="E10:41_4_1_1_3"/>
<font color="Maroon">q1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q2</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E5:41"><i><font color="Green">E39</font></i></a>, </i>;<br/><a NAME="E11:41_4_1_1_3"/><b>then </b><i><font color="Green">E123</font></i>: 
( <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E12:41_4_1_1_3"/><b>then </b><i><font color="Green">E124</font></i>: 
( <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font> )
 <b>by </b><i>, <a class="ref" href="urysohn1.html#T6" target="_self">Th6</a>, <a class="txt" href="urysohn1.html#E6:41"><i><font color="Green">E40</font></i></a>, <a class="txt" href="urysohn1.html#E1:41"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E13:41_4_1_1_3"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">q2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 <b>by </b><i>, <a class="txt" href="urysohn1.html#E5:41"><i><font color="Green">E39</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_4_1_1_3_1_1"/>
( <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> or <font color="Maroon">q2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font> )
 </span><b>by </b><i><a class="ref" href="urysohn1.html#T9" target="_self">Th9</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:41_4_1_1_3_1_1_1"/>
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_4_1_1_3_1_1_1"/>
( <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i>, <a class="txt" href="urysohn1.html#E7:41"><i><font color="Green">E41</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:41_4_1_1_3_1_1_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">q2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">q3</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:41_4_1_1_3_1_1_2"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">q3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font>
 <b>by </b><i/>;<br/><a NAME="E4:41_4_1_1_3_1_1_2"/>
( <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q3</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q3</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="urysohn1.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E5:41_4_1_1_3_1_1_2"/><b>then </b>
( <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i>, , <a class="ref" href="urysohn1.html#T6" target="_self">Th6</a>, <a class="txt" href="urysohn1.html#E6:41"><i><font color="Green">E40</font></i></a>, <a class="txt" href="urysohn1.html#E7:41"><i><font color="Green">E41</font></i></a>, <a class="ref" href="urysohn1.html#D6" target="_self">Def6</a>, <a class="txt" href="urysohn1.html#E1:41"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><a NAME="E6:41_4_1_1_3_1_1_2"/><b>then </b>
( <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E7:41_4_1_1_3_1_1_2"/>
( <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E15:41_4_1_1_3"/>
( <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_1_4"><b>suppose </b></a><a NAME="E1:41_4_1_1_4"/><i><font color="Green">E129</font></i>: 
( not <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> &amp; not <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><a NAME="E2:41_4_1_1_4"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:41_4_1_1_4"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:41_4_1_1_4"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">r1</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">r1</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">q11</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font>, <font color="Maroon">q21</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E6:41_4_1_1_4"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">q11</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">r1</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">q21</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">r1</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E7:41_4_1_1_4"/>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> is    <a href="urysohn1.html#M1" target="_self">Between</a> of <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q11</font>,<font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q21</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E4:41"><i><font color="Green">E32</font></i></a>, <a class="ref" href="urysohn1.html#T5" target="_self">Th5</a>, <a class="txt" href="urysohn1.html#E5:41"><i><font color="Green">E39</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">D01</font> being    <a href="urysohn1.html#M1" target="_self">Between</a> of <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q11</font>,<font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q21</font><b> such that </b><br/><a NAME="E9:41_4_1_1_4"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">D01</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font>
 ;<br/><a NAME="E10:41_4_1_1_4"/>
<font color="Maroon">q11</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q21</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E5:41"><i><font color="Green">E39</font></i></a>, </i>;<br/><a NAME="E11:41_4_1_1_4"/><b>then </b><i><font color="Green">E137</font></i>: 
( <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q11</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q11</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q21</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q11</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q21</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E12:41_4_1_1_4"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E13:41_4_1_1_4"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E14:41_4_1_1_4"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">r2</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">r2</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">q12</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font>, <font color="Maroon">q22</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E16:41_4_1_1_4"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">q12</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">r2</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">q22</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" target="_self">axis</a> <font color="Maroon">r2</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</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> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E17:41_4_1_1_4"/>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font> is    <a href="urysohn1.html#M1" target="_self">Between</a> of <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q12</font>,<font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q22</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E4:41"><i><font color="Green">E32</font></i></a>, , </i>;<br/><b>then consider </b><font color="Maroon">D02</font> being    <a href="urysohn1.html#M1" target="_self">Between</a> of <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q12</font>,<font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q22</font><b> such that </b><br/><a NAME="E19:41_4_1_1_4"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">D02</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r2</font>
 ;<br/><a NAME="E20:41_4_1_1_4"/>
<font color="Maroon">q12</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q22</font>
 <b>by </b><i>, </i>;<br/><a NAME="E21:41_4_1_1_4"/><b>then </b><i><font color="Green">E145</font></i>: 
( <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q12</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q12</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q22</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q12</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q22</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E22:41_4_1_1_4"/>
( <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i>, <a class="ref" href="urysohn1.html#T6" target="_self">Th6</a>, <a class="txt" href="urysohn1.html#E6:41"><i><font color="Green">E40</font></i></a>, <a class="txt" href="urysohn1.html#E7:41"><i><font color="Green">E41</font></i></a>, <a class="ref" href="urysohn1.html#T9" target="_self">Th9</a>, <a class="ref" href="urysohn1.html#D6" target="_self">Def6</a>, <a class="txt" href="urysohn1.html#E1:41"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E23:41_4_1_1_4"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">q21</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q12</font>
 <b>by </b><i>, , <a class="txt" href="urysohn1.html#E5:41"><i><font color="Green">E39</font></i></a>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_4_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_4_1_1_4_1_1"/>
( <font color="Maroon">q21</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q12</font> or <font color="Maroon">q21</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q12</font> )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:41_4_1_1_4_1_1_1"/>
<font color="Maroon">q21</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q12</font>
 ;<br/><div class="add"><a NAME="E2:41_4_1_1_4_1_1_1"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q21</font> &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q21</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q21</font></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q21</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i>, <a class="ref" href="urysohn1.html#T6" target="_self">Th6</a>, <a class="txt" href="urysohn1.html#E6:41"><i><font color="Green">E40</font></i></a>, <a class="txt" href="urysohn1.html#E7:41"><i><font color="Green">E41</font></i></a>, <a class="ref" href="urysohn1.html#T9" target="_self">Th9</a>, <a class="ref" href="urysohn1.html#D6" target="_self">Def6</a>, , <a class="txt" href="urysohn1.html#E1:41"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><a NAME="E3:41_4_1_1_4_1_1_1"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q21</font></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q21</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E4:41_4_1_1_4_1_1_1"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_4_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:41_4_1_1_4_1_1_2"/>
<font color="Maroon">q21</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q12</font>
 ;<br/><div class="add"><a NAME="E2:41_4_1_1_4_1_1_2"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q21</font> &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q21</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q21</font></span>)</span> &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q12</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q12</font></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q21</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q12</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q12</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i>, , <a class="ref" href="urysohn1.html#T6" target="_self">Th6</a>, <a class="txt" href="urysohn1.html#E6:41"><i><font color="Green">E40</font></i></a>, <a class="txt" href="urysohn1.html#E7:41"><i><font color="Green">E41</font></i></a>, <a class="ref" href="urysohn1.html#T9" target="_self">Th9</a>, <a class="ref" href="urysohn1.html#D6" target="_self">Def6</a>, , <a class="txt" href="urysohn1.html#E1:41"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><a NAME="E3:41_4_1_1_4_1_1_2"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q21</font></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q21</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q12</font> &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q12</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E4:41_4_1_1_4_1_1_2"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q12</font> &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">q12</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E5:41_4_1_1_4_1_1_2"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E25:41_4_1_1_4"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:41_4"/>
( <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r2</font> )
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E19:41"/>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1" target="_self">dyadic</a> <font color="Maroon">n</font> implies <font color="Maroon">F</font> <a href="urysohn1.html#K4" target="_self">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> )
 <b>by </b><i/>;<br/>


</div>
