<?xml version="1.0"?>
<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="compts_1.html#V5" title="COMPTS_1:attr.5">being_T4</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</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="E1:15"/><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c4">G</font> be    <a href="urysohn3.html#M2" title="URYSOHN3:mode.2">Rain</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  ) &amp; ( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> ) &amp; ( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  implies for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st $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="Olive" title="b1">n</font> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 ) );<br/>
<a NAME="E2:15"/><i><font color="Green" title="E16">A2</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#K3" title="URYSOHN1:func.3">DOM</a>   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="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> ;<br/>

<a NAME="E1:15_1"/><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 )
 
<b>by </b><i><a class="ref" href="urysohn1.html#D5" title="URYSOHN1:def.5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">a</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <font color="Maroon" title="c7">b</font> = 1 as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E3:15_1"/><i><font color="Green" title="E18">A4</font></i>: 
 <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="urysohn2.html#T32" title="URYSOHN2:th.32">URYSOHN2:32</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 )
 </span><b>by </b><i><a class="txt" href="urysohn3.html#E1:15_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c8">s</font> =  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><a NAME="E3:15_1_1_1"/><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font>
 ;<br/><a NAME="E4:15_1_1_1"/>
( not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 &amp; not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_1_1_1"/><i><font color="Green" title="E21">A7</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1_1_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  )
 </span><b>by </b><i><a class="txt" href="urysohn3.html#E1:15_1_1_1_1"><i><font color="Green" title="E21">A7</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1
 ;<br/><div class="add"><a NAME="E2:15_1_1_1_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="urysohn1.html#T2" title="URYSOHN1:th.2">URYSOHN1:2</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:15_1_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="urysohn1.html#T1" title="URYSOHN1:th.1">URYSOHN1:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_1_1_1_2"/><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/><a NAME="E3:15_1_1_1_1_1_2"/>
( <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="urysohn3.html#E3:15_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="urysohn3.html#E1:15_1_1_1_1_1_2"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="measure5.html#D1" title="MEASURE5:def.1">MEASURE5:def 1</a></i>;<br/><b>hence </b><a NAME="E4:15_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:15_1_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="urysohn1.html#T1" title="URYSOHN1:th.1">URYSOHN1:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E5:15_1_1_1"/>
 ex <font color="Olive" title="b1">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="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]
 <b>by </b><i><a class="txt" href="urysohn3.html#E3:15_1_1_1"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_2"/><i><font color="Green" title="E19">A9</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">s</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="E3:15_1_1_2"/><i><font color="Green" title="E20">A10</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c5">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="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:15"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="urysohn3.html#T10" target="_self" title="URYSOHN3:th.10">Th10</a></i>;<br/><a NAME="E4:15_1_1_2"/><i><font color="Green" title="E21">A11</font></i>: 
not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_1_2_1"/><i><font color="Green" title="E22">A12</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E3:15_1_1_2_1"/>
( <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E3:15_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="urysohn3.html#E1:15_1_1_2"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="measure5.html#D1" title="MEASURE5:def.1">MEASURE5:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:15_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:15_1_1_2_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="urysohn1.html#T2" title="URYSOHN1:th.2">URYSOHN1:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:15_1_1_2"/>
not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_1_2_2"/><i><font color="Green" title="E22">A13</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E3:15_1_1_2_2"/>
( <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E3:15_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="urysohn3.html#E1:15_1_1_2"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="measure5.html#D1" title="MEASURE5:def.1">MEASURE5:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:15_1_1_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:15_1_1_2_2"><i><font color="Green" title="E22">A13</font></i></a>, <a class="ref" href="urysohn1.html#T1" title="URYSOHN1:th.1">URYSOHN1:1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:15_1_1_2"/>
 ex <font color="Olive" title="b1">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="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]
 <b>by </b><i><a class="txt" href="urysohn3.html#E3:15_1_1_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="urysohn3.html#E4:15_1_1_2"><i><font color="Green" title="E21">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_3"><b>suppose </b></a><a NAME="E1:15_1_1_3"/><i><font color="Green" title="E19">A14</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1
 ;<br/><div class="add"><a NAME="E2:15_1_1_3"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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>then reconsider </b><font color="Maroon" title="c8">s</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/><a NAME="E4:15_1_1_3"/><i><font color="Green" title="E20">A15</font></i>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font>
 ;<br/><a NAME="E5:15_1_1_3"/>
( not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1_1_3_1"/><i><font color="Green" title="E21">A16</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_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="E1:15_1_1_3_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  )
 </span><b>by </b><i><a class="txt" href="urysohn3.html#E1:15_1_1_3_1"><i><font color="Green" title="E21">A16</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_3_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:15_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="urysohn1.html#T1" title="URYSOHN1:th.1">URYSOHN1:1</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:15_1_1_3"><i><font color="Green" title="E19">A14</font></i></a>, <a class="ref" href="urysohn1.html#T2" title="URYSOHN1:th.2">URYSOHN1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_3_1_1_2"/><i><font color="Green" title="E22">A17</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/><a NAME="E3:15_1_1_3_1_1_2"/>
( <font color="Maroon" title="c6">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c9">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  )
 <b>by </b><i><a class="txt" href="urysohn3.html#E3:15_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="urysohn3.html#E1:15_1_1_3_1_1_2"><i><font color="Green" title="E22">A17</font></i></a>, <a class="ref" href="measure5.html#D1" title="MEASURE5:def.1">MEASURE5:def 1</a></i>;<br/><b>hence </b><a NAME="E4:15_1_1_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:15_1_1_3"><i><font color="Green" title="E19">A14</font></i></a>, <a class="ref" href="urysohn1.html#T2" title="URYSOHN1:th.2">URYSOHN1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E6:15_1_1_3"/>
 ex <font color="Olive" title="b1">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="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]
 <b>by </b><i><a class="txt" href="urysohn3.html#E4:15_1_1_3"><i><font color="Green" title="E20">A15</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:15"/>
 ex <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> , <a href="pcomps_1.html#K1" title="PCOMPS_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> st <br/>for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">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="urysohn3.html#E2:15"><i><font color="Green" title="E16">A2</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c5">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> , <a href="pcomps_1.html#K1" title="PCOMPS_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="E5:15"/><i><font color="Green" title="E17">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#K3" title="URYSOHN1:func.3">DOM</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b1">x</font>]
 ;<br/>
<b>take </b>
<font color="Maroon" title="c5">F</font>
;<br/>

<b>thus </b><a NAME="E6:15"/>
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a>  holds <br/>( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 implies <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  implies for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Olive" title="b2">n</font> holds <br/><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="urysohn3.html#E5:15"><i><font color="Green" title="E17">A18</font></i></a></i>;<br/>


</div>
