<?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="E1:28"/><i><font color="Green" title="E26">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>let </b><font color="Maroon" title="c5">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E2:28"/><i><font color="Green" title="E27">A2</font></i>: 
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1</span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c6">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E3:28"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">r</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1"/>
( <font color="Maroon" title="c5">r</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">r</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#E2:28"><i><font color="Green" title="E27">A2</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="28_1_1"><b>suppose </b></a><a NAME="E1:28_1_1"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c5">r</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"><a NAME="E2:28_1_1"/><b>then </b>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12" title="PROB_1:func.12">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> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:28_1_1"/><b>then </b><i><font color="Green" title="E30">A5</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> 
 <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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_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="E1:28_1_1_1_1"/>
(  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  or  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1_1"><b>case </b></a><a NAME="E1:28_1_1_1_1_1"/>
 <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:28_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="urysohn3.html#K7" title="URYSOHN3:func.7">.</a> <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:28"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="urysohn3.html#D6" target="_self" title="URYSOHN3:def.6">Def6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1_2"><b>case </b></a><a NAME="E1:28_1_1_1_1_2"/>
 <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c7">S</font> =  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">G</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c8">er</font> = <font color="Maroon" title="c5">r</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="E4:28_1_1_1_1_2"/>
for <font color="Olive" title="b1">r1</font> being <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> holds <br/><font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">er</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">r1</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:28_1_1_1_1_2_1"/><i><font color="Green" title="E31">A6</font></i>: 
<font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 ;<br/>

<a NAME="E2:28_1_1_1_1_2_1"/><i><font color="Green" title="E32">A7</font></i>: 
<font color="Maroon" title="c7">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 
<b>by </b><i><a class="ref" href="urysohn3.html#T15" target="_self" title="URYSOHN3:th.15">Th15</a></i>;<br/>
<a NAME="E3:28_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c9">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:28_1_1_1_1_2_1"><i><font color="Green" title="E31">A6</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">p1</font> = <font color="Maroon" title="c9">r1</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>assume </b><a NAME="E5:28_1_1_1_1_2_1"/><i><font color="Green" title="E33">A8</font></i>: 
not <font color="Maroon" title="c9">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">er</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_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="E1:28_1_1_1_1_2_1_1"/>
(  <a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c10">p1</font> or  <a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c10">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E34">A9</font></i>: 
 <a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c10">p1</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c11">n</font> =  <a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c10">p1</font>;<br/><a NAME="E2:28_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E35">A10</font></i>: 
<font color="Maroon" title="c10">p1</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"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c10">p1</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:28_1_1_1_1_2_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="urysohn3.html#E2:28_1_1_1_1_2_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="urysohn3.html#T6" target="_self" title="URYSOHN3:th.6">Th6</a></i>;<br/><a NAME="E3:28_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c5">r</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"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c10">p1</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:28_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="urysohn3.html#E1:28_1_1_1_1_2_1_1_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="urysohn3.html#T7" target="_self" title="URYSOHN3:th.7">Th7</a></i>;<br/><a NAME="E4:28_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E36">A11</font></i>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">r</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> <span class="p2">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c10">p1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:28"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="urysohn3.html#E1:28_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="urysohn3.html#E3:28_1_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></i>;<br/><a NAME="E5:28_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12" title="PROB_1:func.12">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> 
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:28_1_1_1_1_2_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="urysohn3.html#E2:28_1_1_1_1_2_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E6:28_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> 
 <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/><a NAME="E7:28_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E37">A12</font></i>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">p1</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> <span class="p2">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c10">p1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">p1</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:28"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="urysohn3.html#E1:28_1_1_1_1_2_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="urysohn3.html#E2:28_1_1_1_1_2_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="urysohn3.html#E2:28_1_1_1_1_2_1_1_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">D</font> = <font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c10">p1</font></span>)</span> as    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>, <a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c10">p1</font> <b>by </b><i><a class="txt" href="urysohn3.html#E1:28"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="urysohn3.html#D2" target="_self" title="URYSOHN3:def.2">Def2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">r</font> = <font color="Maroon" title="c5">r</font>, <font color="Maroon" title="c14">p1</font> = <font color="Maroon" title="c10">p1</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"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c10">p1</font></span>)</span> <b>by </b><i><a class="txt" href="urysohn3.html#E1:28_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="urysohn3.html#E1:28_1_1_1_1_2_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="urysohn3.html#E2:28_1_1_1_1_2_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="urysohn3.html#E1:28_1_1_1_1_2_1_1_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="urysohn3.html#T7" target="_self" title="URYSOHN3:th.7">Th7</a></i>;<br/><a NAME="E10:28_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E38">A13</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c14">p1</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:28"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="urysohn3.html#E5:28_1_1_1_1_2_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></i>;<br/><a NAME="E11:28_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c13">r</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="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c13">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/><a NAME="E12:28_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c13">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c14">p1</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E10:28_1_1_1_1_2_1_1_1"><i><font color="Green" title="E38">A13</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E13:28_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E3:28"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="urysohn3.html#E1:28_1_1_1_1_2_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="urysohn3.html#E4:28_1_1_1_1_2_1_1_1"><i><font color="Green" title="E36">A11</font></i></a>, <a class="txt" href="urysohn3.html#E7:28_1_1_1_1_2_1_1_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="ref" href="urysohn3.html#D5" target="_self" title="URYSOHN3:def.5">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E34">A14</font></i>: 
 <a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c10">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">r</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c11">n</font> =  <a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">r</font>;<br/><a NAME="E2:28_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E35">A15</font></i>: 
<font color="Maroon" title="c10">p1</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"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c10">p1</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:28_1_1_1_1_2_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="urysohn3.html#E2:28_1_1_1_1_2_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="urysohn3.html#T6" target="_self" title="URYSOHN3:th.6">Th6</a></i>;<br/><a NAME="E3:28_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E36">A16</font></i>: 
 <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c10">p1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:28_1_1_1_1_2_1_1_2"><i><font color="Green" title="E34">A14</font></i></a>, <a class="ref" href="urysohn2.html#T33" title="URYSOHN2:th.33">URYSOHN2:33</a></i>;<br/><a NAME="E4:28_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c5">r</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"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:28_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="urysohn3.html#T7" target="_self" title="URYSOHN3:th.7">Th7</a></i>;<br/><a NAME="E5:28_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E37">A17</font></i>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">r</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> <span class="p2">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:28"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="urysohn3.html#E1:28_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="urysohn3.html#E3:28_1_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></i>;<br/><a NAME="E6:28_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12" title="PROB_1:func.12">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> 
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:28_1_1_1_1_2_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="urysohn3.html#E2:28_1_1_1_1_2_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:28_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> 
 <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/><a NAME="E8:28_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E38">A18</font></i>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">p1</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> <span class="p2">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">p1</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:28"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="urysohn3.html#E1:28_1_1_1_1_2_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="urysohn3.html#E2:28_1_1_1_1_2_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="urysohn3.html#E2:28_1_1_1_1_2_1_1_2"><i><font color="Green" title="E35">A15</font></i></a>, <a class="txt" href="urysohn3.html#E3:28_1_1_1_1_2_1_1_2"><i><font color="Green" title="E36">A16</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">D</font> = <font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">r</font></span>)</span> as    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>, <a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">r</font> <b>by </b><i><a class="txt" href="urysohn3.html#E1:28"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="urysohn3.html#D2" target="_self" title="URYSOHN3:def.2">Def2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">r</font> = <font color="Maroon" title="c5">r</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"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">r</font></span>)</span> <b>by </b><i><a class="txt" href="urysohn3.html#E1:28_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="urysohn3.html#T7" target="_self" title="URYSOHN3:th.7">Th7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">p1</font> = <font color="Maroon" title="c10">p1</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"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">r</font></span>)</span> <b>by </b><i><a class="txt" href="urysohn3.html#E2:28_1_1_1_1_2_1_1_2"><i><font color="Green" title="E35">A15</font></i></a>, <a class="txt" href="urysohn3.html#E3:28_1_1_1_1_2_1_1_2"><i><font color="Green" title="E36">A16</font></i></a></i>;<br/><a NAME="E12:28_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E39">A19</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c14">p1</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:28"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="urysohn3.html#E5:28_1_1_1_1_2_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></i>;<br/><a NAME="E13:28_1_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c13">r</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="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c13">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/><a NAME="E14:28_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c13">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c14">p1</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E12:28_1_1_1_1_2_1_1_2"><i><font color="Green" title="E39">A19</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E15:28_1_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E3:28"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="urysohn3.html#E1:28_1_1_1_1_2_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="urysohn3.html#E5:28_1_1_1_1_2_1_1_2"><i><font color="Green" title="E37">A17</font></i></a>, <a class="txt" href="urysohn3.html#E8:28_1_1_1_1_2_1_1_2"><i><font color="Green" title="E38">A18</font></i></a>, <a class="ref" href="urysohn3.html#D5" target="_self" title="URYSOHN3:def.5">Def5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:28_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">r</font> is    <a href="supinf_1.html#M1" title="SUPINF_1:mode.1">majorant</a> of <font color="Maroon" title="c7">S</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D9" title="SUPINF_1:def.9">SUPINF_1:def 9</a></i>;<br/><a NAME="E6:28_1_1_1_1_2"/><b>then </b>
 <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">er</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D16" title="SUPINF_1:def.16">SUPINF_1:def 16</a></i>;<br/><b>hence </b><a NAME="E7:28_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="urysohn3.html#K7" title="URYSOHN3:func.7">.</a> <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="ref" href="urysohn3.html#D6" target="_self" title="URYSOHN3:def.6">Def6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:28_1_1"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="urysohn3.html#K7" title="URYSOHN3:func.7">.</a> <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2"><b>suppose </b></a><a NAME="E1:28_1_2"/>
<font color="Maroon" title="c5">r</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="E2:28_1_2"/><b>then </b><i><font color="Green" title="E29">A20</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="ref" href="urysohn1.html#T2" title="URYSOHN1:th.2">URYSOHN1:2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_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="E1:28_1_2_1_1"/>
(  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  or  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2_1_1_1"><b>case </b></a><a NAME="E1:28_1_2_1_1_1"/>
 <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:28_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="urysohn3.html#K7" title="URYSOHN3:func.7">.</a> <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:28_1_2"><i><font color="Green" title="E29">A20</font></i></a>, <a class="ref" href="urysohn3.html#D6" target="_self" title="URYSOHN3:def.6">Def6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2_1_1_2"><b>case </b></a><a NAME="E1:28_1_2_1_1_2"/>
 <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">S</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <b> such that </b><br/><a NAME="E3:28_1_2_1_1_2"/><i><font color="Green" title="E30">A21</font></i>: 
<font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c4">G</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c8">e1</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="E5:28_1_2_1_1_2"/>
<font color="Maroon" title="c8">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E6:28_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E31">A22</font></i>: 
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">S</font> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">e1</font> )
 <b>by </b><i><a class="txt" href="urysohn3.html#E3:28_1_2_1_1_2"><i><font color="Green" title="E30">A21</font></i></a>, <a class="ref" href="urysohn3.html#T17" target="_self" title="URYSOHN3:th.17">Th17</a></i>;<br/><a NAME="E7:28_1_2_1_1_2"/>
( <span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="urysohn3.html#K7" title="URYSOHN3:func.7">.</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c7">S</font> &amp;  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="urysohn3.html#E3:28_1_2_1_1_2"><i><font color="Green" title="E30">A21</font></i></a>, <a class="ref" href="urysohn3.html#D6" target="_self" title="URYSOHN3:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E8:28_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="urysohn3.html#K7" title="URYSOHN3:func.7">.</a> <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:28_1_2"><i><font color="Green" title="E29">A20</font></i></a>, <a class="txt" href="urysohn3.html#E6:28_1_2_1_1_2"><i><font color="Green" title="E31">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:28_1_2"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="urysohn3.html#K7" title="URYSOHN3:func.7">.</a> <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
