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<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c1">T</font> <a href="compts_1.html#NR4" title="COMPTS_1:NR.4">is_T4</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E2:41"/><i><font color="Green" title="E25">A2</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>

<b>let </b><font color="Maroon" title="c4">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c5">G</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E3:41"/><i><font color="Green" title="E26">A3</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E4:41"/><i><font color="Green" title="E27">A4</font></i>: 
for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font>  st <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/>( <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b1">r1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b1">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r2</font> )
 ;<br/>

<a NAME="E5:41"/><i><font color="Green" title="E28">A5</font></i>: 
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="urysohn1.html#E2:41"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E6:41"/><i><font color="Green" title="E29">A6</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> )
 
<b>by </b><i><a class="ref" href="urysohn1.html#T12" target="_self" title="URYSOHN1:th.12">Th12</a></i>;<br/>
<a NAME="E7:41"/><i><font color="Green" title="E30">A7</font></i>: 
for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font>  holds <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="c6">r</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">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"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font> )
 </span><b>by </b><i><a class="ref" href="urysohn1.html#T5" target="_self" title="URYSOHN1:th.5">Th5</a></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"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_1_1"/>
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c6">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="urysohn1.html#E3:41"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="urysohn1.html#E5:41"><i><font color="Green" title="E28">A5</font></i></a></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" title="E31">A8</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c7">q1</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> <b>by </b><i><a class="ref" href="urysohn1.html#T12" target="_self" title="URYSOHN1:th.12">Th12</a></i>;<br/><a NAME="E3:41_1_1_2"/>
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c7">q1</font> &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c7">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c7">q1</font></span>)</span> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c7">q1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c6">r</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E3:41"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="urysohn1.html#E4:41"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="urysohn1.html#E1:41_1_1_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/><a NAME="E4:41_1_1_2"/><b>then </b>
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c7">q1</font></span>)</span> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c7">q1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c6">r</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:41_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c6">r</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E6:41_1_1_2"/>
<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c6">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="urysohn1.html#E5:41"><i><font color="Green" title="E28">A5</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c6">S</font> = <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font></span>)</span> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="urysohn1.html#T15" target="_self" title="URYSOHN1:th.15">Th15</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">S</font>,  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>] <b>means </b>for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> holds <br/>for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font>  st <font color="Olive" title="b2">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Olive" title="b1">r</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Olive" title="b3">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Olive" title="b1">r</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> holds <br/>$2 is    <a href="urysohn1.html#M1" title="URYSOHN1:mode.1">Between</a> of <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r1</font>,<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b3">r2</font>;<br/>
<a NAME="E9:41"/><i><font color="Green" title="E31">A9</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">S</font>  ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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" title="c7">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">S</font>;<br/>

<a NAME="E1:41_2"/><i><font color="Green" title="E32">A10</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">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" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c8">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c8">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> )
 
<b>by </b><i><a class="txt" href="urysohn1.html#E1:41_2"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="urysohn1.html#T20" target="_self" title="URYSOHN1:th.20">Th20</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">q1</font>, <font color="Maroon" title="c10">q2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font><b> such that </b><br/><a NAME="E5:41_2"/><i><font color="Green" title="E33">A11</font></i>: 
<font color="Maroon" title="c9">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c8">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:41_2"/><i><font color="Green" title="E34">A12</font></i>: 
<font color="Maroon" title="c10">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c8">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ;<br/>
<b>consider </b><font color="Maroon" title="c11">Q0</font> being    <a href="urysohn1.html#M1" title="URYSOHN1:mode.1">Between</a> of <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">q1</font>,<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">q2</font>;<br/>
<b>take </b>
<font color="Maroon" title="c11">Q0</font>
;<br/>

<b>thus </b><a NAME="E8:41_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c11">Q0</font>]
 <b>by </b><i><a class="txt" href="urysohn1.html#E5:41_2"><i><font color="Green" title="E33">A11</font></i></a>, <a class="txt" href="urysohn1.html#E6:41_2"><i><font color="Green" title="E34">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c7">D</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c6">S</font>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E11:41"/><i><font color="Green" title="E32">A13</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">S</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c7">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="urysohn1.html#E9:41"><i><font color="Green" title="E31">A9</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>] <b>means </b>for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> holds <br/>( ( <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> ) &amp; ( not <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> ) );<br/>
<a NAME="E12:41"/><i><font color="Green" title="E33">A14</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>  ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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" title="c8">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> or  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">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" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> ;<br/><b>consider </b><font color="Maroon" title="c10">y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:41_3_1_1"/><i><font color="Green" title="E34">A15</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">x</font>
 ;<br/><b>take </b>
<font color="Maroon" title="c10">y</font>
;<br/><b>thus </b><a NAME="E5:41_3_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">y</font>]
 <b>by </b><i><a class="txt" href="urysohn1.html#E4:41_3_1_1"><i><font color="Green" title="E34">A15</font></i></a></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" title="E34">A16</font></i>: 
not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c6">S</font> <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">y</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:41_3_1_2"/><i><font color="Green" title="E35">A17</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font>
 ;<br/><b>take </b>
<font color="Maroon" title="c10">y</font>
;<br/><b>thus </b><a NAME="E5:41_3_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">y</font>]
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:41_3_1_2"><i><font color="Green" title="E34">A16</font></i></a>, <a class="txt" href="urysohn1.html#E4:41_3_1_2"><i><font color="Green" title="E35">A17</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E14:41"/><i><font color="Green" title="E34">A18</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="urysohn1.html#E12:41"><i><font color="Green" title="E33">A14</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c8">F</font>
;<br/>

<a NAME="E15:41"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="urysohn1.html#T12" target="_self" title="URYSOHN1:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E16:41"/>
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E3:41"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="urysohn1.html#E6:41"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="urysohn1.html#E14:41"><i><font color="Green" title="E34">A18</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c9">r1</font>, <font color="Maroon" title="c10">r2</font>, <font color="Maroon" title="c11">r</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>

<b>assume </b><a NAME="E17:41"/><i><font color="Green" title="E35">A19</font></i>: 
<font color="Maroon" title="c9">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r2</font>
 ;<br/>

<b>thus </b><a NAME="E18:41"/>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">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" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> ) or ( <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> &amp; not <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> ) or ( not <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> ) or ( not <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> &amp; not <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">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" title="E36">A20</font></i>: 
( <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> )
 ;<br/><div class="add"><a NAME="E2:41_4_1_1_1"/><b>then </b><i><font color="Green" title="E37">A21</font></i>: 
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r1</font> &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r2</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E14:41"><i><font color="Green" title="E34">A18</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">r1</font> = <font color="Maroon" title="c9">r1</font>, <font color="Maroon" title="c13">r2</font> = <font color="Maroon" title="c10">r2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> <b>by </b><i><a class="txt" href="urysohn1.html#E1:41_4_1_1_1"><i><font color="Green" title="E36">A20</font></i></a></i>;<br/><a NAME="E4:41_4_1_1_1"/>
<font color="Maroon" title="c12">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">r2</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E17:41"><i><font color="Green" title="E35">A19</font></i></a></i>;<br/><b>hence </b><a NAME="E5:41_4_1_1_1"/>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E4:41"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="urysohn1.html#E2:41_4_1_1_1"><i><font color="Green" title="E37">A21</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"><b>suppose </b></a><a NAME="E1:41_4_1_1_2"/><i><font color="Green" title="E36">A22</font></i>: 
( <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> &amp; not <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> )
 ;<br/><div class="add"><a NAME="E2:41_4_1_1_2"/><b>then </b><i><font color="Green" title="E37">A23</font></i>: 
<font color="Maroon" title="c10">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:41_4_1_1_2"/><i><font color="Green" title="E38">A24</font></i>: 
<font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r2</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E14:41"><i><font color="Green" title="E34">A18</font></i></a>, <a class="txt" href="urysohn1.html#E1:41_4_1_1_2"><i><font color="Green" title="E36">A22</font></i></a></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" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c10">r2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c10">r2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:41_4_1_1_2"><i><font color="Green" title="E36">A22</font></i></a>, <a class="ref" href="urysohn1.html#T20" target="_self" title="URYSOHN1:th.20">Th20</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">q1</font>, <font color="Maroon" title="c13">q2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font><b> such that </b><br/><a NAME="E6:41_4_1_1_2"/><i><font color="Green" title="E39">A25</font></i>: 
( <font color="Maroon" title="c12">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c10">r2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c10">r2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E7:41_4_1_1_2"/>
<font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r2</font> is    <a href="urysohn1.html#M1" title="URYSOHN1:mode.1">Between</a> of <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font>,<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E11:41"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="urysohn1.html#E2:41_4_1_1_2"><i><font color="Green" title="E37">A23</font></i></a>, <a class="txt" href="urysohn1.html#E6:41_4_1_1_2"><i><font color="Green" title="E39">A25</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">D0</font> being    <a href="urysohn1.html#M1" title="URYSOHN1:mode.1">Between</a> of <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font>,<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font><b> such that </b><br/><a NAME="E9:41_4_1_1_2"/><i><font color="Green" title="E40">A26</font></i>: 
<font color="Maroon" title="c14">D0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r2</font>
 ;<br/><a NAME="E10:41_4_1_1_2"/>
<font color="Maroon" title="c12">q1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">q2</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E6:41_4_1_1_2"><i><font color="Green" title="E39">A25</font></i></a>, <a class="ref" href="urysohn1.html#T18" target="_self" title="URYSOHN1:th.18">Th18</a></i>;<br/><a NAME="E11:41_4_1_1_2"/><b>then </b><i><font color="Green" title="E41">A27</font></i>: 
( <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E4:41"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="urysohn1.html#E7:41"><i><font color="Green" title="E30">A7</font></i></a></i>;<br/><a NAME="E12:41_4_1_1_2"/><b>then </b><i><font color="Green" title="E42">A28</font></i>: 
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:41"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="urysohn1.html#E3:41_4_1_1_2"><i><font color="Green" title="E38">A24</font></i></a>, <a class="txt" href="urysohn1.html#E9:41_4_1_1_2"><i><font color="Green" title="E40">A26</font></i></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a></i>;<br/><a NAME="E13:41_4_1_1_2"/><i><font color="Green" title="E43">A29</font></i>: 
<font color="Maroon" title="c9">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">q1</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E17:41"><i><font color="Green" title="E35">A19</font></i></a>, <a class="txt" href="urysohn1.html#E6:41_4_1_1_2"><i><font color="Green" title="E39">A25</font></i></a>, <a class="ref" href="urysohn1.html#T22" target="_self" title="URYSOHN1:th.22">Th22</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" title="c9">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q1</font> or <font color="Maroon" title="c9">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q1</font> )
 </span><b>by </b><i><a class="txt" href="urysohn1.html#E13:41_4_1_1_2"><i><font color="Green" title="E43">A29</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">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" title="c9">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_4_1_1_2_1_1_1"/>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E14:41"><i><font color="Green" title="E34">A18</font></i></a>, <a class="txt" href="urysohn1.html#E11:41_4_1_1_2"><i><font color="Green" title="E41">A27</font></i></a>, <a class="txt" href="urysohn1.html#E12:41_4_1_1_2"><i><font color="Green" title="E42">A28</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" title="E44">A30</font></i>: 
<font color="Maroon" title="c9">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">q1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c15">q0</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font><b> such that </b><br/><a NAME="E3:41_4_1_1_2_1_1_2"/><i><font color="Green" title="E45">A31</font></i>: 
<font color="Maroon" title="c15">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">r1</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:41_4_1_1_2"><i><font color="Green" title="E36">A22</font></i></a></i>;<br/><a NAME="E4:41_4_1_1_2_1_1_2"/>
( <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q0</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E3:41"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="urysohn1.html#E4:41"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="urysohn1.html#E1:41_4_1_1_2_1_1_2"><i><font color="Green" title="E44">A30</font></i></a>, <a class="txt" href="urysohn1.html#E3:41_4_1_1_2_1_1_2"><i><font color="Green" title="E45">A31</font></i></a></i>;<br/><a NAME="E5:41_4_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font> &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font></span>)</span> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:41"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="urysohn1.html#E14:41"><i><font color="Green" title="E34">A18</font></i></a>, <a class="txt" href="urysohn1.html#E3:41_4_1_1_2"><i><font color="Green" title="E38">A24</font></i></a>, <a class="txt" href="urysohn1.html#E9:41_4_1_1_2"><i><font color="Green" title="E40">A26</font></i></a>, <a class="txt" href="urysohn1.html#E11:41_4_1_1_2"><i><font color="Green" title="E41">A27</font></i></a>, <a class="txt" href="urysohn1.html#E3:41_4_1_1_2_1_1_2"><i><font color="Green" title="E45">A31</font></i></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a>, <a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/><a NAME="E6:41_4_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font></span>)</span> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E7:41_4_1_1_2_1_1_2"/>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">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" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">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" title="E36">A32</font></i>: 
( not <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> )
 ;<br/><div class="add"><a NAME="E2:41_4_1_1_3"/><b>then </b><i><font color="Green" title="E37">A33</font></i>: 
<font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:41_4_1_1_3"/><i><font color="Green" title="E38">A34</font></i>: 
<font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r1</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E14:41"><i><font color="Green" title="E34">A18</font></i></a>, <a class="txt" href="urysohn1.html#E1:41_4_1_1_3"><i><font color="Green" title="E36">A32</font></i></a></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" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c9">r1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c9">r1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:41_4_1_1_3"><i><font color="Green" title="E36">A32</font></i></a>, <a class="ref" href="urysohn1.html#T20" target="_self" title="URYSOHN1:th.20">Th20</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">q1</font>, <font color="Maroon" title="c13">q2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font><b> such that </b><br/><a NAME="E6:41_4_1_1_3"/><i><font color="Green" title="E39">A35</font></i>: 
( <font color="Maroon" title="c12">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c9">r1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c9">r1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E7:41_4_1_1_3"/>
<font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r1</font> is    <a href="urysohn1.html#M1" title="URYSOHN1:mode.1">Between</a> of <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font>,<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E11:41"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="urysohn1.html#E2:41_4_1_1_3"><i><font color="Green" title="E37">A33</font></i></a>, <a class="txt" href="urysohn1.html#E6:41_4_1_1_3"><i><font color="Green" title="E39">A35</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">D0</font> being    <a href="urysohn1.html#M1" title="URYSOHN1:mode.1">Between</a> of <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font>,<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font><b> such that </b><br/><a NAME="E9:41_4_1_1_3"/><i><font color="Green" title="E40">A36</font></i>: 
<font color="Maroon" title="c14">D0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r1</font>
 ;<br/><a NAME="E10:41_4_1_1_3"/>
<font color="Maroon" title="c12">q1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">q2</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E6:41_4_1_1_3"><i><font color="Green" title="E39">A35</font></i></a>, <a class="ref" href="urysohn1.html#T18" target="_self" title="URYSOHN1:th.18">Th18</a></i>;<br/><a NAME="E11:41_4_1_1_3"/><b>then </b><i><font color="Green" title="E41">A37</font></i>: 
( <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E4:41"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="urysohn1.html#E7:41"><i><font color="Green" title="E30">A7</font></i></a></i>;<br/><a NAME="E12:41_4_1_1_3"/><b>then </b><i><font color="Green" title="E42">A38</font></i>: 
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:41"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="urysohn1.html#E3:41_4_1_1_3"><i><font color="Green" title="E38">A34</font></i></a>, <a class="txt" href="urysohn1.html#E9:41_4_1_1_3"><i><font color="Green" title="E40">A36</font></i></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a></i>;<br/><a NAME="E13:41_4_1_1_3"/><i><font color="Green" title="E43">A39</font></i>: 
<font color="Maroon" title="c13">q2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">r2</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E17:41"><i><font color="Green" title="E35">A19</font></i></a>, <a class="txt" href="urysohn1.html#E6:41_4_1_1_3"><i><font color="Green" title="E39">A35</font></i></a>, <a class="ref" href="urysohn1.html#T22" target="_self" title="URYSOHN1:th.22">Th22</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" title="c13">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r2</font> or <font color="Maroon" title="c13">q2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r2</font> )
 </span><b>by </b><i><a class="txt" href="urysohn1.html#E13:41_4_1_1_3"><i><font color="Green" title="E43">A39</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">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" title="c13">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_4_1_1_3_1_1_1"/>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E14:41"><i><font color="Green" title="E34">A18</font></i></a>, <a class="txt" href="urysohn1.html#E11:41_4_1_1_3"><i><font color="Green" title="E41">A37</font></i></a>, <a class="txt" href="urysohn1.html#E12:41_4_1_1_3"><i><font color="Green" title="E42">A38</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" title="E44">A40</font></i>: 
<font color="Maroon" title="c13">q2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">r2</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c15">q3</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font><b> such that </b><br/><a NAME="E3:41_4_1_1_3_1_1_2"/><i><font color="Green" title="E45">A41</font></i>: 
<font color="Maroon" title="c15">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r2</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:41_4_1_1_3"><i><font color="Green" title="E36">A32</font></i></a></i>;<br/><a NAME="E4:41_4_1_1_3_1_1_2"/>
( <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q3</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q3</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E3:41"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="urysohn1.html#E4:41"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="urysohn1.html#E1:41_4_1_1_3_1_1_2"><i><font color="Green" title="E44">A40</font></i></a>, <a class="txt" href="urysohn1.html#E3:41_4_1_1_3_1_1_2"><i><font color="Green" title="E45">A41</font></i></a></i>;<br/><a NAME="E5:41_4_1_1_3_1_1_2"/><b>then </b>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font> &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font></span>)</span> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:41"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="urysohn1.html#E14:41"><i><font color="Green" title="E34">A18</font></i></a>, <a class="txt" href="urysohn1.html#E3:41_4_1_1_3"><i><font color="Green" title="E38">A34</font></i></a>, <a class="txt" href="urysohn1.html#E9:41_4_1_1_3"><i><font color="Green" title="E40">A36</font></i></a>, <a class="txt" href="urysohn1.html#E11:41_4_1_1_3"><i><font color="Green" title="E41">A37</font></i></a>, <a class="txt" href="urysohn1.html#E3:41_4_1_1_3_1_1_2"><i><font color="Green" title="E45">A41</font></i></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a>, <a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/><a NAME="E6:41_4_1_1_3_1_1_2"/><b>then </b>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font></span>)</span> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E7:41_4_1_1_3_1_1_2"/>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">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" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">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" title="E36">A42</font></i>: 
( not <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> &amp; not <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> )
 ;<br/><div class="add"><a NAME="E2:41_4_1_1_4"/><b>then </b><i><font color="Green" title="E37">A43</font></i>: 
<font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:41_4_1_1_4"/><i><font color="Green" title="E38">A44</font></i>: 
<font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r1</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E14:41"><i><font color="Green" title="E34">A18</font></i></a>, <a class="txt" href="urysohn1.html#E1:41_4_1_1_4"><i><font color="Green" title="E36">A42</font></i></a></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" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c9">r1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c9">r1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:41_4_1_1_4"><i><font color="Green" title="E36">A42</font></i></a>, <a class="ref" href="urysohn1.html#T20" target="_self" title="URYSOHN1:th.20">Th20</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">q11</font>, <font color="Maroon" title="c13">q21</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font><b> such that </b><br/><a NAME="E6:41_4_1_1_4"/><i><font color="Green" title="E39">A45</font></i>: 
( <font color="Maroon" title="c12">q11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c9">r1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c13">q21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c9">r1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E7:41_4_1_1_4"/>
<font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r1</font> is    <a href="urysohn1.html#M1" title="URYSOHN1:mode.1">Between</a> of <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q11</font>,<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E11:41"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="urysohn1.html#E2:41_4_1_1_4"><i><font color="Green" title="E37">A43</font></i></a>, <a class="txt" href="urysohn1.html#E6:41_4_1_1_4"><i><font color="Green" title="E39">A45</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c14">D01</font> being    <a href="urysohn1.html#M1" title="URYSOHN1:mode.1">Between</a> of <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q11</font>,<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font><b> such that </b><br/><a NAME="E9:41_4_1_1_4"/><i><font color="Green" title="E40">A46</font></i>: 
<font color="Maroon" title="c14">D01</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">r1</font>
 ;<br/><a NAME="E10:41_4_1_1_4"/>
<font color="Maroon" title="c12">q11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">q21</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E6:41_4_1_1_4"><i><font color="Green" title="E39">A45</font></i></a>, <a class="ref" href="urysohn1.html#T18" target="_self" title="URYSOHN1:th.18">Th18</a></i>;<br/><a NAME="E11:41_4_1_1_4"/><b>then </b><i><font color="Green" title="E41">A47</font></i>: 
( <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q11</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q11</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c12">q11</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E4:41"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="urysohn1.html#E7:41"><i><font color="Green" title="E30">A7</font></i></a></i>;<br/><a NAME="E12:41_4_1_1_4"/><i><font color="Green" title="E42">A48</font></i>: 
<font color="Maroon" title="c10">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:41_4_1_1_4"><i><font color="Green" title="E36">A42</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E13:41_4_1_1_4"/><i><font color="Green" title="E43">A49</font></i>: 
<font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r2</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E14:41"><i><font color="Green" title="E34">A18</font></i></a>, <a class="txt" href="urysohn1.html#E1:41_4_1_1_4"><i><font color="Green" title="E36">A42</font></i></a></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" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c10">r2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c10">r2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:41_4_1_1_4"><i><font color="Green" title="E36">A42</font></i></a>, <a class="ref" href="urysohn1.html#T20" target="_self" title="URYSOHN1:th.20">Th20</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">q12</font>, <font color="Maroon" title="c16">q22</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font><b> such that </b><br/><a NAME="E16:41_4_1_1_4"/><i><font color="Green" title="E44">A50</font></i>: 
( <font color="Maroon" title="c15">q12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c10">r2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c16">q22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn1.html#K6" title="URYSOHN1:func.6">axis</a> <font color="Maroon" title="c10">r2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E17:41_4_1_1_4"/>
<font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r2</font> is    <a href="urysohn1.html#M1" title="URYSOHN1:mode.1">Between</a> of <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q12</font>,<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c16">q22</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E11:41"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="urysohn1.html#E12:41_4_1_1_4"><i><font color="Green" title="E42">A48</font></i></a>, <a class="txt" href="urysohn1.html#E16:41_4_1_1_4"><i><font color="Green" title="E44">A50</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">D02</font> being    <a href="urysohn1.html#M1" title="URYSOHN1:mode.1">Between</a> of <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q12</font>,<font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c16">q22</font><b> such that </b><br/><a NAME="E19:41_4_1_1_4"/><i><font color="Green" title="E45">A51</font></i>: 
<font color="Maroon" title="c17">D02</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r2</font>
 ;<br/><a NAME="E20:41_4_1_1_4"/>
<font color="Maroon" title="c15">q12</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">q22</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E16:41_4_1_1_4"><i><font color="Green" title="E44">A50</font></i></a>, <a class="ref" href="urysohn1.html#T18" target="_self" title="URYSOHN1:th.18">Th18</a></i>;<br/><a NAME="E21:41_4_1_1_4"/><b>then </b><i><font color="Green" title="E46">A52</font></i>: 
( <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q12</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q12</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c16">q22</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q12</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c16">q22</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E4:41"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="urysohn1.html#E7:41"><i><font color="Green" title="E30">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E22:41_4_1_1_4"/>
( <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:41"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="urysohn1.html#E3:41_4_1_1_4"><i><font color="Green" title="E38">A44</font></i></a>, <a class="txt" href="urysohn1.html#E9:41_4_1_1_4"><i><font color="Green" title="E40">A46</font></i></a>, <a class="txt" href="urysohn1.html#E11:41_4_1_1_4"><i><font color="Green" title="E41">A47</font></i></a>, <a class="txt" href="urysohn1.html#E13:41_4_1_1_4"><i><font color="Green" title="E43">A49</font></i></a>, <a class="txt" href="urysohn1.html#E19:41_4_1_1_4"><i><font color="Green" title="E45">A51</font></i></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a></i>;<br/><a NAME="E23:41_4_1_1_4"/><i><font color="Green" title="E47">A53</font></i>: 
<font color="Maroon" title="c13">q21</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">q12</font>
 <b>by </b><i><a class="txt" href="urysohn1.html#E17:41"><i><font color="Green" title="E35">A19</font></i></a>, <a class="txt" href="urysohn1.html#E1:41_4_1_1_4"><i><font color="Green" title="E36">A42</font></i></a>, <a class="txt" href="urysohn1.html#E6:41_4_1_1_4"><i><font color="Green" title="E39">A45</font></i></a>, <a class="txt" href="urysohn1.html#E16:41_4_1_1_4"><i><font color="Green" title="E44">A50</font></i></a>, <a class="ref" href="urysohn1.html#T23" target="_self" title="URYSOHN1:th.23">Th23</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" title="c13">q21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q12</font> or <font color="Maroon" title="c13">q21</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">q12</font> )
 </span><b>by </b><i><a class="txt" href="urysohn1.html#E23:41_4_1_1_4"><i><font color="Green" title="E47">A53</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">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" title="c13">q21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">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#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font> &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font></span>)</span> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:41"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="urysohn1.html#E3:41_4_1_1_4"><i><font color="Green" title="E38">A44</font></i></a>, <a class="txt" href="urysohn1.html#E9:41_4_1_1_4"><i><font color="Green" title="E40">A46</font></i></a>, <a class="txt" href="urysohn1.html#E11:41_4_1_1_4"><i><font color="Green" title="E41">A47</font></i></a>, <a class="txt" href="urysohn1.html#E13:41_4_1_1_4"><i><font color="Green" title="E43">A49</font></i></a>, <a class="txt" href="urysohn1.html#E19:41_4_1_1_4"><i><font color="Green" title="E45">A51</font></i></a>, <a class="txt" href="urysohn1.html#E21:41_4_1_1_4"><i><font color="Green" title="E46">A52</font></i></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a>, <a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">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#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font></span>)</span> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">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#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">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" title="c13">q21</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">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#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font> &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font></span>)</span> &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q12</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q12</font></span>)</span> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q12</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q12</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E1:41"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="urysohn1.html#E4:41"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="urysohn1.html#E3:41_4_1_1_4"><i><font color="Green" title="E38">A44</font></i></a>, <a class="txt" href="urysohn1.html#E9:41_4_1_1_4"><i><font color="Green" title="E40">A46</font></i></a>, <a class="txt" href="urysohn1.html#E11:41_4_1_1_4"><i><font color="Green" title="E41">A47</font></i></a>, <a class="txt" href="urysohn1.html#E13:41_4_1_1_4"><i><font color="Green" title="E43">A49</font></i></a>, <a class="txt" href="urysohn1.html#E19:41_4_1_1_4"><i><font color="Green" title="E45">A51</font></i></a>, <a class="txt" href="urysohn1.html#E21:41_4_1_1_4"><i><font color="Green" title="E46">A52</font></i></a>, <a class="ref" href="urysohn1.html#D10" target="_self" title="URYSOHN1:def.10">Def10</a>, <a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">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#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font></span>)</span> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c13">q21</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q12</font> &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q12</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">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#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q12</font> &amp; <font color="Maroon" title="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c15">q12</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">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#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">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#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">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" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c9">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c10">r2</font> )
 ;<br/>


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

<b>thus </b><a NAME="E19:41"/>
( <font color="Maroon" title="c11">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c4">n</font> implies <font color="Maroon" title="c8">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c11">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">r</font> )
 <b>by </b><i><a class="txt" href="urysohn1.html#E14:41"><i><font color="Green" title="E34">A18</font></i></a></i>;<br/>


</div>
