<?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="E1:29"/><i><font color="Green" title="E28">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>reconsider </b><font color="Maroon" title="c5">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="E3:29"/><i><font color="Green" title="E29">A2</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="urysohn1.html#T6" title="URYSOHN1:th.6">URYSOHN1:6</a></i>;<br/>
<a NAME="E4:29"/><b>then </b><i><font color="Green" title="E30">A3</font></i>: 
1 <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="ref" href="urysohn1.html#D4" title="URYSOHN1:def.4">URYSOHN1:def 4</a></i>;<br/>
<a NAME="E5:29"/><b>then </b>
1 <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> 
 
<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="E6:29"/><b>then </b><i><font color="Green" title="E31">A4</font></i>: 
1 <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="E7:29"/><i><font color="Green" title="E32">A5</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="urysohn1.html#T6" title="URYSOHN1:th.6">URYSOHN1:6</a></i>;<br/>
<a NAME="E8:29"/><b>then </b><i><font color="Green" title="E33">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#K2" title="URYSOHN1:func.2">DYADIC</a> 
 
<b>by </b><i><a class="ref" href="urysohn1.html#D4" title="URYSOHN1:def.4">URYSOHN1:def 4</a></i>;<br/>
<a NAME="E9:29"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> 
 
<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="E10:29"/><b>then </b><i><font color="Green" title="E34">A7</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#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="E11:29"/><i><font color="Green" title="E35">A8</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds <br/> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( <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> implies <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> implies <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_1"/>
(  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</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">x</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="29_1_1_1_1"><b>case </b></a><a NAME="E1:29_1_1_1_1"/><i><font color="Green" title="E36">A9</font></i>: 
 <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</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"><a NAME="E2:29_1_1_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> implies <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_1_1_1"/><i><font color="Green" title="E37">A10</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 ;<br/>

<a NAME="E2:29_1_1_1_1_1"/><i><font color="Green" title="E38">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> 1 <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="urysohn3.html#E3:29"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="urysohn3.html#E4:29"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="urysohn3.html#E6:29"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></i>;<br/>
<a NAME="E3:29_1_1_1_1_1"/>
( <font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is    <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="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> ) )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="urysohn3.html#D2" target="_self" title="URYSOHN3:def.2">Def2</a></i>;<br/>
<a NAME="E4:29_1_1_1_1_1"/><b>then </b>
<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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></i>;<br/>
<a NAME="E5:29_1_1_1_1_1"/><b>then </b>
for <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> not <font color="Maroon" title="c6">x</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="Olive" title="b1">s</font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29_1_1_1_1_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="urysohn3.html#E2:29_1_1_1_1_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:29_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:29_1_1_1_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="urysohn3.html#D5" target="_self" title="URYSOHN3:def.5">Def5</a>, <a class="ref" href="urysohn2.html#T31" title="URYSOHN2:th.31">URYSOHN2:31</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:29_1_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> &amp; <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> implies <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> implies <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:29_1_1_1_1"><i><font color="Green" title="E36">A9</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="29_1_1_1_2"><b>case </b></a><a NAME="E1:29_1_1_1_2"/><i><font color="Green" title="E36">A12</font></i>: 
 <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</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">x</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/><a NAME="E3:29_1_1_1_2"/><i><font color="Green" title="E37">A13</font></i>: 
<font color="Maroon" title="c5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E4:29_1_1_1_2"/><b>then </b><i><font color="Green" title="E38">A14</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_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c7">S</font> &amp;  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">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="c5">e1</font> )
 <b>by </b><i><a class="ref" href="urysohn3.html#T17" target="_self" title="URYSOHN3:th.17">Th17</a></i>;<br/><a NAME="E5:29_1_1_1_2"/><i><font color="Green" title="E39">A15</font></i>: 
( <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">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="ref" href="urysohn3.html#D6" target="_self" title="URYSOHN3:def.6">Def6</a></i>;<br/><a NAME="E6:29_1_1_1_2"/><i><font color="Green" title="E40">A16</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> implies <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_1_2_1"/><i><font color="Green" title="E41">A17</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<a NAME="E2:29_1_1_1_2_1"/><i><font color="Green" title="E42">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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="urysohn3.html#E7:29"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="urysohn3.html#E8:29"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="urysohn3.html#E10:29"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></i>;<br/>
<a NAME="E3:29_1_1_1_2_1"/>
( <font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is    <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="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> ) )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="urysohn3.html#D2" target="_self" title="URYSOHN3:def.2">Def2</a></i>;<br/>
<a NAME="E4:29_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E43">A19</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></i>;<br/>
<a NAME="E5:29_1_1_1_2_1"/><i><font color="Green" title="E44">A20</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>  st  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/> not <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c4">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">s</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:29_1_1_1_2_1_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">s</font>
 ;<br/>

<b>assume </b><a NAME="E2:29_1_1_1_2_1_1"/><i><font color="Green" title="E45">A21</font></i>: 
<font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c4">G</font>
 ;<br/>

<a NAME="E3:29_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E46">A22</font></i>: 
( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font> holds <br/> not <font color="Maroon" title="c6">x</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="Olive" title="b1">t</font> ) )
 
<b>by </b><i><a class="ref" href="urysohn3.html#D5" target="_self" title="URYSOHN3:def.5">Def5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">s</font> = <font color="Maroon" title="c8">s</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E5:29_1_1_1_2_1_1"/>
<font color="Maroon" title="c9">s</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> 
 
<b>by </b><i><a class="txt" href="urysohn3.html#E3:29_1_1_1_2_1_1"><i><font color="Green" title="E46">A22</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="E6:29_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E47">A23</font></i>: 
<font color="Maroon" title="c9">s</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/>
<b>consider </b><font color="Maroon" title="c10">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> <b> such that </b><br/><a NAME="E8:29_1_1_1_2_1_1"/><i><font color="Green" title="E48">A24</font></i>: 
<font color="Maroon" title="c9">s</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="c10">n</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E3:29_1_1_1_2_1_1"><i><font color="Green" title="E46">A22</font></i></a>, <a class="ref" href="urysohn1.html#D4" title="URYSOHN1:def.4">URYSOHN1:def 4</a></i>;<br/>
<a NAME="E9:29_1_1_1_2_1_1"/><i><font color="Green" title="E49">A25</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="c9">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="Maroon" title="c10">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="urysohn3.html#E3:29_1_1_1_2_1_1"><i><font color="Green" title="E46">A22</font></i></a>, <a class="txt" href="urysohn3.html#E6:29_1_1_1_2_1_1"><i><font color="Green" title="E47">A23</font></i></a>, <a class="txt" href="urysohn3.html#E8:29_1_1_1_2_1_1"><i><font color="Green" title="E48">A24</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="c11">D</font> = <font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c10">n</font> 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>,<font color="Maroon" title="c10">n</font> <b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">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="c12">r1</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> , <font color="Maroon" title="c13">r2</font> = <font color="Maroon" title="c9">s</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="c10">n</font> <b>by </b><i><a class="txt" href="urysohn3.html#E8:29_1_1_1_2_1_1"><i><font color="Green" title="E48">A24</font></i></a>, <a class="ref" href="urysohn1.html#T12" title="URYSOHN1:th.12">URYSOHN1:12</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <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="c9">s</font> )
 </span><b>by </b><i><a class="txt" href="urysohn3.html#E8:29_1_1_1_2_1_1"><i><font color="Green" title="E48">A24</font></i></a>, <a class="ref" href="urysohn1.html#T5" title="URYSOHN1:th.5">URYSOHN1:5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1_2_1_1_1_1"/><i><font color="Green" title="E50">A26</font></i>: 
<font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:29_1_1_1_2_1"><i><font color="Green" title="E41">A17</font></i></a>, <a class="txt" href="urysohn3.html#E2:29_1_1_1_2_1_1"><i><font color="Green" title="E45">A21</font></i></a>, <a class="txt" href="urysohn3.html#E9:29_1_1_1_2_1_1"><i><font color="Green" title="E49">A25</font></i></a>, <a class="txt" href="urysohn3.html#E1:29_1_1_1_2_1_1_1_1"><i><font color="Green" title="E50">A26</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="29_1_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_1_2_1_1_1_2"/>
 <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="c9">s</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E50">A27</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="c11">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c12">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c13">r2</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></i>;<br/><a NAME="E3:29_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c11">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c12">r1</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="c11">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c12">r1</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="E4:29_1_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c12">r1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c13">r2</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:29_1_1_1_2_1_1_1_2"><i><font color="Green" title="E50">A27</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:29_1_1_1_2_1_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="c11">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:29_1_1_1_2_1_1_1_2"><i><font color="Green" title="E50">A27</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="E6:29_1_1_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:29_1_1_1_2_1"><i><font color="Green" title="E41">A17</font></i></a>, <a class="txt" href="urysohn3.html#E2:29_1_1_1_2_1_1"><i><font color="Green" title="E45">A21</font></i></a>, <a class="txt" href="urysohn3.html#E9:29_1_1_1_2_1_1"><i><font color="Green" title="E49">A25</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>
<b>consider </b><font color="Maroon" title="c8">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:29_1_1_1_2_1"/><i><font color="Green" title="E45">A28</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c4">G</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:29_1_1_1_2"><i><font color="Green" title="E36">A12</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="E8:29_1_1_1_2_1"/><i><font color="Green" title="E46">A29</font></i>: 
<font color="Maroon" title="c8">a</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#E7:29_1_1_1_2_1"><i><font color="Green" title="E45">A28</font></i></a>, <a class="ref" href="urysohn3.html#D5" target="_self" title="URYSOHN3:def.5">Def5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c8">a</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>consider </b><font color="Maroon" title="c10">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> <b> such that </b><br/><a NAME="E11:29_1_1_1_2_1"/><i><font color="Green" title="E47">A30</font></i>: 
<font color="Maroon" title="c9">a</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="c10">n</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E8:29_1_1_1_2_1"><i><font color="Green" title="E46">A29</font></i></a>, <a class="ref" href="urysohn1.html#D4" title="URYSOHN1:def.4">URYSOHN1:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_2_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1_1_2_1_2"/>
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <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="c9">a</font> )
 </span><b>by </b><i><a class="txt" href="urysohn3.html#E11:29_1_1_1_2_1"><i><font color="Green" title="E47">A30</font></i></a>, <a class="ref" href="urysohn1.html#T5" title="URYSOHN1:th.5">URYSOHN1:5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_2_1_2_1"><b>suppose </b></a><a NAME="E1:29_1_1_1_2_1_2_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_1_1_1_2_1_2_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <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#E1:29_1_1_1_2_1"><i><font color="Green" title="E41">A17</font></i></a>, <a class="txt" href="urysohn3.html#E2:29_1_1_1_2_1"><i><font color="Green" title="E42">A18</font></i></a>, <a class="txt" href="urysohn3.html#E4:29_1_1_1_2_1"><i><font color="Green" title="E43">A19</font></i></a>, <a class="txt" href="urysohn3.html#E7:29_1_1_1_2_1"><i><font color="Green" title="E45">A28</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="29_1_1_1_2_1_2_2"><b>suppose </b></a><a NAME="E1:29_1_1_1_2_1_2_2"/>
 <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="c9">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_1_1_1_2_1_2_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <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#E5:29_1_1_1_2_1"><i><font color="Green" title="E44">A20</font></i></a>, <a class="txt" href="urysohn3.html#E7:29_1_1_1_2_1"><i><font color="Green" title="E45">A28</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E7:29_1_1_1_2"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> implies <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_1_1_2_2"/><i><font color="Green" title="E41">A31</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 ;<br/>

<a NAME="E2:29_1_1_1_2_2"/><i><font color="Green" title="E42">A32</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> 1 <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="urysohn3.html#E3:29"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="urysohn3.html#E4:29"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="urysohn3.html#E6:29"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></i>;<br/>
<a NAME="E3:29_1_1_1_2_2"/>
( <font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is    <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="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> ) )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="urysohn3.html#D2" target="_self" title="URYSOHN3:def.2">Def2</a></i>;<br/>
<a NAME="E4:29_1_1_1_2_2"/><b>then </b>
<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"><span class="p2">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></i>;<br/>
<a NAME="E5:29_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E43">A33</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> not <font color="Maroon" title="c6">x</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="Olive" title="b1">s</font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29_1_1_1_2_2"><i><font color="Green" title="E41">A31</font></i></a>, <a class="txt" href="urysohn3.html#E2:29_1_1_1_2_2"><i><font color="Green" title="E42">A32</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="E6:29_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E44">A34</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c4">G</font>
 
<b>by </b><i><a class="ref" href="urysohn3.html#D5" target="_self" title="URYSOHN3:def.5">Def5</a>, <a class="ref" href="urysohn2.html#T31" title="URYSOHN2:th.31">URYSOHN2:31</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">S</font> =  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c6">x</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>  <b>by </b><i><a class="txt" href="urysohn3.html#E5:29_1_1_1_2_2"><i><font color="Green" title="E43">A33</font></i></a>, <a class="ref" href="urysohn3.html#D5" target="_self" title="URYSOHN3:def.5">Def5</a>, <a class="ref" href="urysohn2.html#T31" title="URYSOHN2:th.31">URYSOHN2:31</a></i>;<br/>
<a NAME="E8:29_1_1_1_2_2"/><i><font color="Green" title="E45">A35</font></i>: 
<font color="Maroon" title="c5">e1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c8">S</font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E6:29_1_1_1_2_2"><i><font color="Green" title="E44">A34</font></i></a>, <a class="ref" href="supinf_1.html#T76" title="SUPINF_1:th.76">SUPINF_1:76</a></i>;<br/>
<a NAME="E9:29_1_1_1_2_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K6" title="SUPINF_1:func.6">sup</a> <font color="Maroon" title="c8">S</font>
 
<b>by </b><i><a class="ref" href="urysohn3.html#D6" target="_self" title="URYSOHN3:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E10:29_1_1_1_2_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="urysohn3.html#E4:29_1_1_1_2"><i><font color="Green" title="E38">A14</font></i></a>, <a class="txt" href="urysohn3.html#E8:29_1_1_1_2_2"><i><font color="Green" title="E45">A35</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:29_1_1_1_2"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> &amp; <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> implies <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> implies <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 <b>by </b><i><a class="txt" href="urysohn3.html#E3:29_1_1_1_2"><i><font color="Green" title="E37">A13</font></i></a>, <a class="txt" href="urysohn3.html#E5:29_1_1_1_2"><i><font color="Green" title="E39">A15</font></i></a>, <a class="txt" href="urysohn3.html#E6:29_1_1_1_2"><i><font color="Green" title="E40">A16</font></i></a>, <a class="ref" href="urysohn3.html#T17" target="_self" title="URYSOHN3:th.17">Th17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:29_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> &amp; <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> implies <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> implies <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:29"/>
 <a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c4">G</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<a NAME="E1:29_2"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds   <a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c4">G</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><b>proof </b></a><div class="add">

<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/>

<a NAME="E1:29_2_1"/>
for <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>   st <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> holds <br/> ex <font color="Olive" title="b2">H</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b2">H</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H</font> &amp; <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="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:29_2_1_1"/><i><font color="Green" title="E36">A36</font></i>: 
( <font color="Maroon" title="c7">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">Q</font> = <font color="Maroon" title="c7">Q</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E3:29_2_1_1"/><i><font color="Green" title="E37">A37</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <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 <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  &amp; the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  )
 
<b>by </b><i><a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E4:29_2_1_1"/><i><font color="Green" title="E38">A38</font></i>: 
<font color="Maroon" title="c8">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29_2_1_1"><i><font color="Green" title="E36">A36</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">x</font> = <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="txt" href="urysohn3.html#E1:29_2_1_1"><i><font color="Green" title="E36">A36</font></i></a>, <a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:29_2_1_1"/><i><font color="Green" title="E39">A39</font></i>: 
( <font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Q</font> )
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:29_2_1_1"><i><font color="Green" title="E36">A36</font></i></a>, <a class="txt" href="urysohn3.html#E3:29_2_1_1"><i><font color="Green" title="E37">A37</font></i></a>, <a class="txt" href="urysohn3.html#E4:29_2_1_1"><i><font color="Green" title="E38">A38</font></i></a>, <a class="ref" href="pcomps_1.html#D5" title="PCOMPS_1:def.5">PCOMPS_1:def 5</a></i>;<br/>
<a NAME="E8:29_2_1_1"/>
 ex <font color="Olive" title="b1">r0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font> &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="Olive" title="b1">r0</font> &amp; <font color="Olive" title="b1">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_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="E1:29_2_1_1_1_1"/>
( <font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 or 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_2_1_1_1_1_1"/><i><font color="Green" title="E40">A40</font></i>: 
<font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c11">r0</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:29_2_1_1_1_1_1"/><i><font color="Green" title="E41">A41</font></i>: 
(  <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="c11">r0</font> &amp; <font color="Maroon" title="c11">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font> )
 <b>by </b><i><a class="txt" href="urysohn3.html#E7:29_2_1_1"><i><font color="Green" title="E39">A39</font></i></a>, <a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">r0</font> = <font color="Maroon" title="c11">r0</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c12">r0</font>
;<br/><b>thus </b><a NAME="E5:29_2_1_1_1_1_1"/>
( <font color="Maroon" title="c12">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font> &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="c12">r0</font> &amp; <font color="Maroon" title="c12">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:29_2_1_1_1_1_1"><i><font color="Green" title="E40">A40</font></i></a>, <a class="txt" href="urysohn3.html#E3:29_2_1_1_1_1_1"><i><font color="Green" title="E41">A41</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_2_1_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_2_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">r0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font> &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="Olive" title="b1">r0</font> &amp; <font color="Olive" title="b1">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c11">r0</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E10:29_2_1_1"/><i><font color="Green" title="E40">A42</font></i>: 
( <font color="Maroon" title="c11">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font> &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="c11">r0</font> &amp; <font color="Maroon" title="c11">r0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>
<b>consider </b><font color="Maroon" title="c12">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E12:29_2_1_1"/><i><font color="Green" title="E41">A43</font></i>: 
( <font color="Maroon" title="c12">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  &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="c12">r1</font> &amp; <font color="Maroon" title="c12">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c11">r0</font> )
 <b>by </b><i><a class="txt" href="urysohn3.html#E10:29_2_1_1"><i><font color="Green" title="E40">A42</font></i></a>, <a class="ref" href="urysohn2.html#T28" title="URYSOHN2:th.28">URYSOHN2:28</a></i>;<br/>
<a NAME="E13:29_2_1_1"/><i><font color="Green" title="E42">A44</font></i>: 
<font color="Maroon" title="c12">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E10:29_2_1_1"><i><font color="Green" title="E40">A42</font></i></a>, <a class="txt" href="urysohn3.html#E12:29_2_1_1"><i><font color="Green" title="E41">A43</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E14:29_2_1_1"/><i><font color="Green" title="E43">A45</font></i>: 
( <font color="Maroon" title="c12">r1</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="c12">r1</font> )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E12:29_2_1_1"><i><font color="Green" title="E41">A43</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="E15:29_2_1_1"/><i><font color="Green" title="E44">A46</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">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="c12">r1</font> holds <br/><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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">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="E1:29_2_1_1_2"/>
<font color="Maroon" title="c13">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="c12">r1</font>
 ;<br/>

<a NAME="E2:29_2_1_1_2"/><b>then </b>
<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">r1</font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="urysohn3.html#E14:29_2_1_1"><i><font color="Green" title="E43">A45</font></i></a>, <a class="ref" href="urysohn3.html#T19" target="_self" title="URYSOHN3:th.19">Th19</a></i>;<br/>
<b>hence </b><a NAME="E3:29_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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c13">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E13:29_2_1_1"><i><font color="Green" title="E42">A44</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>
<b>consider </b><font color="Maroon" title="c13">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> <b> such that </b><br/><a NAME="E17:29_2_1_1"/><i><font color="Green" title="E45">A47</font></i>: 
<font color="Maroon" title="c12">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="c13">n</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E12:29_2_1_1"><i><font color="Green" title="E41">A43</font></i></a>, <a class="ref" href="urysohn1.html#D4" title="URYSOHN1:def.4">URYSOHN1:def 4</a></i>;<br/>
<a NAME="E18:29_2_1_1"/>
<font color="Maroon" title="c12">r1</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> 
 
<b>by </b><i><a class="txt" href="urysohn3.html#E12:29_2_1_1"><i><font color="Green" title="E41">A43</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="E19:29_2_1_1"/><b>then </b><i><font color="Green" title="E46">A48</font></i>: 
<font color="Maroon" title="c12">r1</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="E20:29_2_1_1"/><b>then </b><i><font color="Green" title="E47">A49</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="c12">r1</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="Maroon" title="c13">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">r1</font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="urysohn3.html#E12:29_2_1_1"><i><font color="Green" title="E41">A43</font></i></a>, <a class="txt" href="urysohn3.html#E17:29_2_1_1"><i><font color="Green" title="E45">A47</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="c14">D</font> = <font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c13">n</font> 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>,<font color="Maroon" title="c13">n</font> <b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="urysohn3.html#D2" target="_self" title="URYSOHN3:def.2">Def2</a></i>;<br/>
<a NAME="E22:29_2_1_1"/><i><font color="Green" title="E48">A50</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="c13">n</font> &amp; <font color="Maroon" title="c12">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="c13">n</font> &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="c12">r1</font> )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E12:29_2_1_1"><i><font color="Green" title="E41">A43</font></i></a>, <a class="txt" href="urysohn3.html#E17:29_2_1_1"><i><font color="Green" title="E45">A47</font></i></a>, <a class="ref" href="urysohn1.html#T12" title="URYSOHN1:th.12">URYSOHN1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">r1</font> = <font color="Maroon" title="c12">r1</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="c13">n</font> <b>by </b><i><a class="txt" href="urysohn3.html#E17:29_2_1_1"><i><font color="Green" title="E45">A47</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">H</font> = <font color="Maroon" title="c14">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c15">r1</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="E25:29_2_1_1"/>
 ex <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">H</font> &amp; <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="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_2_1_1_3_1"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:29_2_1_1_3_1_1"/><i><font color="Green" title="E49">A51</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c16">H</font>
;<br/><a NAME="E2:29_2_1_1_3_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="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c16">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_1_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_2_1_1_3_1_1_1"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c16">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_2_1_1_3_1_1_1"/><i><font color="Green" title="E50">A52</font></i>: 
( <font color="Maroon" title="c18">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <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> &amp; <font color="Maroon" title="c18">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">H</font> &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c18">x1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">x1</font> = <font color="Maroon" title="c18">x1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="urysohn3.html#E3:29_2_1_1_3_1_1_1"><i><font color="Green" title="E50">A52</font></i></a></i>;<br/>
<a NAME="E5:29_2_1_1_3_1_1_1"/><i><font color="Green" title="E51">A53</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">x1</font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E11:29"><i><font color="Green" title="E35">A8</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">p</font> = <font color="Maroon" title="c9">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c21">y</font> = <font color="Maroon" title="c17">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="txt" href="urysohn3.html#E3:29_2_1_1"><i><font color="Green" title="E37">A37</font></i></a>, <a class="txt" href="urysohn3.html#E3:29_2_1_1_3_1_1_1"><i><font color="Green" title="E50">A52</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">q</font> = <font color="Maroon" title="c21">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="metric_1.html#D14" title="METRIC_1:def.14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E9:29_2_1_1_3_1_1_1"/><i><font color="Green" title="E52">A54</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c21">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15" title="TOPMETR:th.15">TOPMETR:15</a></i>;<br/>
<a NAME="E10:29_2_1_1_3_1_1_1"/><i><font color="Green" title="E53">A55</font></i>: 
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c21">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font> implies <font color="Maroon" title="c21">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">r</font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E11:29_2_1_1_3_1_1_1"/>
<font color="Maroon" title="c22">q</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c20">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="urysohn3.html#E15:29_2_1_1"><i><font color="Green" title="E44">A46</font></i></a>, <a class="txt" href="urysohn3.html#E20:29_2_1_1"><i><font color="Green" title="E47">A49</font></i></a>, <a class="txt" href="urysohn3.html#E1:29_2_1_1_3_1_1"><i><font color="Green" title="E49">A51</font></i></a>, <a class="txt" href="urysohn3.html#E3:29_2_1_1_3_1_1_1"><i><font color="Green" title="E50">A52</font></i></a></i>;<br/>
<a NAME="E12:29_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E54">A56</font></i>: 
( <font color="Maroon" title="c20">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font> &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">q</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font> )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29_2_1_1_3_1_1"><i><font color="Green" title="E49">A51</font></i></a>, <a class="txt" href="urysohn3.html#E3:29_2_1_1_3_1_1_1"><i><font color="Green" title="E50">A52</font></i></a>, <a class="txt" href="urysohn3.html#E5:29_2_1_1_3_1_1_1"><i><font color="Green" title="E51">A53</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E13:29_2_1_1_3_1_1_1"/>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">q</font>
 
;<br/>
<a NAME="E14:29_2_1_1_3_1_1_1"/><b>then </b>
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c20">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">q</font> &amp; <font color="Maroon" title="c20">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font> )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E12:29_2_1_1_3_1_1_1"><i><font color="Green" title="E54">A56</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<a NAME="E15:29_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E55">A57</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">q</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">r</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a></i>;<br/>
<a NAME="E16:29_2_1_1_3_1_1_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">q</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_2_1_1_3_1_1_1_1"/><i><font color="Green" title="E56">A58</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1_3_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="E1:29_2_1_1_3_1_1_1_1_1"/>
(  <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="c20">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">q</font> or <font color="Maroon" title="c20">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">q</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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_2_1_1_3_1_1_1_1_1_1"/>
 <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="c20">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">q</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:29_2_1_1_3_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E12:29_2_1_1_3_1_1_1"><i><font color="Green" title="E54">A56</font></i></a>, <a class="txt" href="urysohn3.html#E1:29_2_1_1_3_1_1_1_1"><i><font color="Green" title="E56">A58</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_2_1_1_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c20">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">q</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> 
 ;<br/><div class="add"><a NAME="E2:29_2_1_1_3_1_1_1_1_1_2"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E3:29_2_1_1_3_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E15:29_2_1_1"><i><font color="Green" title="E44">A46</font></i></a>, <a class="txt" href="urysohn3.html#E20:29_2_1_1"><i><font color="Green" title="E47">A49</font></i></a>, <a class="txt" href="urysohn3.html#E1:29_2_1_1_3_1_1"><i><font color="Green" title="E49">A51</font></i></a>, <a class="txt" href="urysohn3.html#E3:29_2_1_1_3_1_1_1"><i><font color="Green" title="E50">A52</font></i></a>, <a class="txt" href="urysohn3.html#E1:29_2_1_1_3_1_1_1_1"><i><font color="Green" title="E56">A58</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E17:29_2_1_1_3_1_1_1"/>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E7:29_2_1_1"><i><font color="Green" title="E39">A39</font></i></a>, <a class="txt" href="urysohn3.html#E9:29_2_1_1_3_1_1_1"><i><font color="Green" title="E52">A54</font></i></a>, <a class="txt" href="urysohn3.html#E10:29_2_1_1_3_1_1_1"><i><font color="Green" title="E53">A55</font></i></a>, <a class="txt" href="urysohn3.html#E15:29_2_1_1_3_1_1_1"><i><font color="Green" title="E55">A57</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:29_2_1_1_3_1_1"/>
( <font color="Maroon" title="c16">H</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">H</font> &amp; <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="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c16">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Q</font> )
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="urysohn3.html#E19:29_2_1_1"><i><font color="Green" title="E46">A48</font></i></a>, <a class="txt" href="urysohn3.html#E20:29_2_1_1"><i><font color="Green" title="E47">A49</font></i></a>, <a class="txt" href="urysohn3.html#E22:29_2_1_1"><i><font color="Green" title="E48">A50</font></i></a>, <a class="txt" href="urysohn3.html#E1:29_2_1_1_3_1_1"><i><font color="Green" title="E49">A51</font></i></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a>, <a class="ref" href="urysohn3.html#T18" target="_self" title="URYSOHN3:th.18">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:29_2_1_1_3_1_2"/><i><font color="Green" title="E49">A59</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:29_2_1_1_3_1_2"/><i><font color="Green" title="E50">A60</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">p</font> &amp; <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="urysohn3.html#E11:29"><i><font color="Green" title="E35">A8</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">x</font> = <font color="Maroon" title="c9">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><b>consider </b><font color="Maroon" title="c18">r1</font>, <font color="Maroon" title="c19">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:29_2_1_1_3_1_2"/><i><font color="Green" title="E51">A61</font></i>: 
( <font color="Maroon" title="c18">r1</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; <font color="Maroon" title="c19">r2</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="c18">r1</font> &amp; <font color="Maroon" title="c18">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">x</font> &amp; <font color="Maroon" title="c17">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">r2</font> &amp; <font color="Maroon" title="c19">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font> )
 <b>by </b><i><a class="txt" href="urysohn3.html#E7:29_2_1_1"><i><font color="Green" title="E39">A39</font></i></a>, <a class="txt" href="urysohn3.html#E1:29_2_1_1_3_1_2"><i><font color="Green" title="E49">A59</font></i></a>, <a class="txt" href="urysohn3.html#E2:29_2_1_1_3_1_2"><i><font color="Green" title="E50">A60</font></i></a>, <a class="ref" href="urysohn2.html#T35" title="URYSOHN2:th.35">URYSOHN2:35</a></i>;<br/><b>consider </b><font color="Maroon" title="c20">r4</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:29_2_1_1_3_1_2"/><i><font color="Green" title="E52">A62</font></i>: 
( <font color="Maroon" title="c20">r4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  &amp; <font color="Maroon" title="c18">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c20">r4</font> &amp; <font color="Maroon" title="c20">r4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c17">x</font> )
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:29_2_1_1_3_1_2"><i><font color="Green" title="E50">A60</font></i></a>, <a class="txt" href="urysohn3.html#E5:29_2_1_1_3_1_2"><i><font color="Green" title="E51">A61</font></i></a>, <a class="ref" href="urysohn2.html#T28" title="URYSOHN2:th.28">URYSOHN2:28</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">r3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E9:29_2_1_1_3_1_2"/><i><font color="Green" title="E53">A63</font></i>: 
( <font color="Maroon" title="c21">r3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a>  &amp; <font color="Maroon" title="c17">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c21">r3</font> &amp; <font color="Maroon" title="c21">r3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">r2</font> )
 <b>by </b><i><a class="txt" href="urysohn3.html#E5:29_2_1_1_3_1_2"><i><font color="Green" title="E51">A61</font></i></a>, <a class="ref" href="urysohn2.html#T29" title="URYSOHN2:th.29">URYSOHN2:29</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">A</font> = <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="c21">r3</font> 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="txt" href="urysohn3.html#E9:29_2_1_1_3_1_2"><i><font color="Green" title="E53">A63</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><a NAME="E11:29_2_1_1_3_1_2"/><i><font color="Green" title="E54">A64</font></i>: 
<font color="Maroon" title="c22">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="urysohn3.html#E9:29_2_1_1_3_1_2"><i><font color="Green" title="E53">A63</font></i></a>, <a class="ref" href="urysohn3.html#T12" target="_self" title="URYSOHN3:th.12">Th12</a></i>;<br/><a NAME="E12:29_2_1_1_3_1_2"/>
( <font color="Maroon" title="c18">r1</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="c18">r1</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="txt" href="urysohn3.html#E5:29_2_1_1_3_1_2"><i><font color="Green" title="E51">A61</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="E13:29_2_1_1_3_1_2"/><b>then </b>
( <font color="Maroon" title="c18">r1</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="c18">r1</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="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E14:29_2_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E55">A65</font></i>: 
<font color="Maroon" title="c18">r1</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/><b>then reconsider </b><font color="Maroon" title="c23">B</font> = <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="c18">r1</font> 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="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">C</font> = <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"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c23">B</font></span>)</span> 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="E17:29_2_1_1_3_1_2"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c23">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c23">B</font>
 ;<br/><a NAME="E18:29_2_1_1_3_1_2"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c23">B</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/><a NAME="E19:29_2_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E56">A66</font></i>: 
<font color="Maroon" title="c24">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6" title="PRE_TOPC:def.6">PRE_TOPC:def 6</a></i>;<br/><b>take </b><font color="Maroon" title="c25">H</font> = <font color="Maroon" title="c22">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c24">C</font>;<br/><a NAME="E20:29_2_1_1_3_1_2"/><i><font color="Green" title="E57">A67</font></i>: 
<font color="Maroon" title="c20">r4</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> 
 <b>by </b><i><a class="txt" href="urysohn3.html#E7:29_2_1_1_3_1_2"><i><font color="Green" title="E52">A62</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="E21:29_2_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c20">r4</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="E22:29_2_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E58">A68</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c23">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c20">r4</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="urysohn3.html#E7:29_2_1_1_3_1_2"><i><font color="Green" title="E52">A62</font></i></a>, <a class="txt" href="urysohn3.html#E14:29_2_1_1_3_1_2"><i><font color="Green" title="E55">A65</font></i></a>, <a class="ref" href="urysohn3.html#T13" target="_self" title="URYSOHN3:th.13">Th13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">r4</font> = <font color="Maroon" title="c20">r4</font> as    <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>  <b>by </b><i><a class="txt" href="urysohn3.html#E20:29_2_1_1_3_1_2"><i><font color="Green" title="E57">A67</font></i></a>, <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="E24:29_2_1_1_3_1_2"/>
not <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="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c26">r4</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="urysohn3.html#E5:29_2_1_1_3_1_2"><i><font color="Green" title="E51">A61</font></i></a>, <a class="txt" href="urysohn3.html#E7:29_2_1_1_3_1_2"><i><font color="Green" title="E52">A62</font></i></a>, <a class="ref" href="urysohn3.html#T20" target="_self" title="URYSOHN3:th.20">Th20</a></i>;<br/><a NAME="E25:29_2_1_1_3_1_2"/><b>then </b>
not <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c23">B</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E22:29_2_1_1_3_1_2"><i><font color="Green" title="E58">A68</font></i></a></i>;<br/><a NAME="E26:29_2_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E59">A69</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">C</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="E27:29_2_1_1_3_1_2"/><i><font color="Green" title="E60">A70</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="c21">r3</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="urysohn3.html#E9:29_2_1_1_3_1_2"><i><font color="Green" title="E53">A63</font></i></a>, <a class="ref" href="urysohn3.html#T18" target="_self" title="URYSOHN3:th.18">Th18</a></i>;<br/><a NAME="E28:29_2_1_1_3_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="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c25">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_1_3_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_2_1_1_3_1_2_1"/>
<font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c25">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c28">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_2_1_1_3_1_2_1"/><i><font color="Green" title="E61">A71</font></i>: 
( <font color="Maroon" title="c28">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <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> &amp; <font color="Maroon" title="c28">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">H</font> &amp; <font color="Maroon" title="c27">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c28">x1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c29">x1</font> = <font color="Maroon" title="c28">x1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="urysohn3.html#E3:29_2_1_1_3_1_2_1"><i><font color="Green" title="E61">A71</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c30">y</font> = <font color="Maroon" title="c27">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="urysohn3.html#E3:29_2_1_1_3_1_2_1"><i><font color="Green" title="E61">A71</font></i></a></i>;<br/>
<a NAME="E6:29_2_1_1_3_1_2_1"/><i><font color="Green" title="E62">A72</font></i>: 
( <font color="Maroon" title="c29">x1</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="c21">r3</font> &amp; <font color="Maroon" title="c29">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c23">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E3:29_2_1_1_3_1_2_1"><i><font color="Green" title="E61">A71</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:29_2_1_1_3_1_2_1"/>
not <font color="Maroon" title="c29">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_1_3_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_2_1_1_3_1_2_1_1"/><i><font color="Green" title="E63">A73</font></i>: 
<font color="Maroon" title="c29">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">B</font>
 ;<br/>

<a NAME="E2:29_2_1_1_3_1_2_1_1"/>
<font color="Maroon" title="c23">B</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> <font color="Maroon" title="c23">B</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<b>hence </b><a NAME="E3:29_2_1_1_3_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E6:29_2_1_1_3_1_2_1"><i><font color="Green" title="E62">A72</font></i></a>, <a class="txt" href="urysohn3.html#E1:29_2_1_1_3_1_2_1_1"><i><font color="Green" title="E63">A73</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:29_2_1_1_3_1_2_1"/><b>then </b><i><font color="Green" title="E63">A74</font></i>: 
<font color="Maroon" title="c18">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x1</font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="urysohn3.html#E14:29_2_1_1_3_1_2"><i><font color="Green" title="E55">A65</font></i></a>, <a class="ref" href="urysohn3.html#T18" target="_self" title="URYSOHN3:th.18">Th18</a></i>;<br/>
<a NAME="E9:29_2_1_1_3_1_2_1"/>
( ( <font color="Maroon" title="c21">r3</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="c21">r3</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="c21">r3</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="txt" href="urysohn3.html#E11:29"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="urysohn3.html#E9:29_2_1_1_3_1_2"><i><font color="Green" title="E53">A63</font></i></a>, <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="E10:29_2_1_1_3_1_2_1"/><b>then </b>
( ( <font color="Maroon" title="c21">r3</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="c21">r3</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="c21">r3</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="c21">r3</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="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E11:29_2_1_1_3_1_2_1"/><b>then </b>
( <font color="Maroon" title="c21">r3</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="c21">r3</font> )
 
<b>by </b><i><a class="ref" href="urysohn1.html#T1" title="URYSOHN1:th.1">URYSOHN1:1</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E12:29_2_1_1_3_1_2_1"/><b>then </b>
<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">r3</font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:29"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="urysohn3.html#E6:29_2_1_1_3_1_2_1"><i><font color="Green" title="E62">A72</font></i></a>, <a class="ref" href="urysohn3.html#T19" target="_self" title="URYSOHN3:th.19">Th19</a></i>;<br/>
<a NAME="E13:29_2_1_1_3_1_2_1"/><b>then </b><i><font color="Green" title="E64">A75</font></i>: 
<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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c29">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">r2</font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E9:29_2_1_1_3_1_2"><i><font color="Green" title="E53">A63</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c31">x</font> = <font color="Maroon" title="c17">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c32">p</font> = <font color="Maroon" title="c31">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c33">y</font> = <font color="Maroon" title="c30">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  <b>by </b><i><a class="txt" href="urysohn3.html#E3:29_2_1_1"><i><font color="Green" title="E37">A37</font></i></a>, <a class="txt" href="urysohn3.html#E3:29_2_1_1_3_1_2_1"><i><font color="Green" title="E61">A71</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c34">q</font> = <font color="Maroon" title="c33">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E18:29_2_1_1_3_1_2_1"/><i><font color="Green" title="E65">A76</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c31">x</font>,<font color="Maroon" title="c33">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15" title="TOPMETR:th.15">TOPMETR:15</a></i>;<br/>
<a NAME="E19:29_2_1_1_3_1_2_1"/><i><font color="Green" title="E66">A77</font></i>: 
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c31">x</font>,<font color="Maroon" title="c33">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font> implies <font color="Maroon" title="c33">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c31">x</font>,<font color="Maroon" title="c10">r</font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E20:29_2_1_1_3_1_2_1"/><i><font color="Green" title="E67">A78</font></i>: 
<font color="Maroon" title="c34">q</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c32">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">r1</font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E5:29_2_1_1_3_1_2"><i><font color="Green" title="E51">A61</font></i></a>, <a class="txt" href="urysohn3.html#E3:29_2_1_1_3_1_2_1"><i><font color="Green" title="E61">A71</font></i></a>, <a class="txt" href="urysohn3.html#E13:29_2_1_1_3_1_2_1"><i><font color="Green" title="E64">A75</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E21:29_2_1_1_3_1_2_1"/><i><font color="Green" title="E68">A79</font></i>: 
<font color="Maroon" title="c32">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c19">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c18">r1</font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E5:29_2_1_1_3_1_2"><i><font color="Green" title="E51">A61</font></i></a>, <a class="txt" href="urysohn3.html#E3:29_2_1_1_3_1_2_1"><i><font color="Green" title="E61">A71</font></i></a>, <a class="txt" href="urysohn3.html#E8:29_2_1_1_3_1_2_1"><i><font color="Green" title="E63">A74</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E22:29_2_1_1_3_1_2_1"/><b>then </b><i><font color="Green" title="E69">A80</font></i>: 
( <font color="Maroon" title="c32">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font> &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">q</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font> )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E5:29_2_1_1_3_1_2"><i><font color="Green" title="E51">A61</font></i></a>, <a class="txt" href="urysohn3.html#E20:29_2_1_1_3_1_2_1"><i><font color="Green" title="E67">A78</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E23:29_2_1_1_3_1_2_1"/>
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c32">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">q</font>
 
;<br/>
<a NAME="E24:29_2_1_1_3_1_2_1"/><b>then </b>
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c10">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c32">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">q</font> &amp; <font color="Maroon" title="c32">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c10">r</font> )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E22:29_2_1_1_3_1_2_1"><i><font color="Green" title="E69">A80</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<a NAME="E25:29_2_1_1_3_1_2_1"/><b>then </b><i><font color="Green" title="E70">A81</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">q</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">r</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T12" title="ABSVALUE:th.12">ABSVALUE:12</a></i>;<br/>
<a NAME="E26:29_2_1_1_3_1_2_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">q</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_1_3_1_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_2_1_1_3_1_2_1_2"/><i><font color="Green" title="E71">A82</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1_3_1_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_2_1_1_3_1_2_1_2_1"/>
(  <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="c32">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">q</font> or <font color="Maroon" title="c32">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">q</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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1_3_1_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:29_2_1_1_3_1_2_1_2_1_1"/>
 <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="c32">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">q</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_2_1_1_3_1_2_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E5:29_2_1_1_3_1_2"><i><font color="Green" title="E51">A61</font></i></a>, <a class="txt" href="urysohn3.html#E21:29_2_1_1_3_1_2_1"><i><font color="Green" title="E68">A79</font></i></a>, <a class="txt" href="urysohn3.html#E1:29_2_1_1_3_1_2_1_2"><i><font color="Green" title="E71">A82</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1_3_1_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:29_2_1_1_3_1_2_1_2_1_2"/>
<font color="Maroon" title="c32">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">q</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> 
 ;<br/><div class="add"><a NAME="E2:29_2_1_1_3_1_2_1_2_1_2"/><b>then </b>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">p</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c34">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E3:29_2_1_1_3_1_2_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E5:29_2_1_1_3_1_2"><i><font color="Green" title="E51">A61</font></i></a>, <a class="txt" href="urysohn3.html#E20:29_2_1_1_3_1_2_1"><i><font color="Green" title="E67">A78</font></i></a>, <a class="txt" href="urysohn3.html#E1:29_2_1_1_3_1_2_1_2"><i><font color="Green" title="E71">A82</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E27:29_2_1_1_3_1_2_1"/>
<font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E7:29_2_1_1"><i><font color="Green" title="E39">A39</font></i></a>, <a class="txt" href="urysohn3.html#E18:29_2_1_1_3_1_2_1"><i><font color="Green" title="E65">A76</font></i></a>, <a class="txt" href="urysohn3.html#E19:29_2_1_1_3_1_2_1"><i><font color="Green" title="E66">A77</font></i></a>, <a class="txt" href="urysohn3.html#E25:29_2_1_1_3_1_2_1"><i><font color="Green" title="E70">A81</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E29:29_2_1_1_3_1_2"/>
( <font color="Maroon" title="c25">H</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">H</font> &amp; <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="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c25">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Q</font> )
 <b>by </b><i><a class="txt" href="urysohn3.html#E11:29_2_1_1_3_1_2"><i><font color="Green" title="E54">A64</font></i></a>, <a class="txt" href="urysohn3.html#E19:29_2_1_1_3_1_2"><i><font color="Green" title="E56">A66</font></i></a>, <a class="txt" href="urysohn3.html#E26:29_2_1_1_3_1_2"><i><font color="Green" title="E59">A69</font></i></a>, <a class="txt" href="urysohn3.html#E27:29_2_1_1_3_1_2"><i><font color="Green" title="E60">A70</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E26:29_2_1_1"/>
 ex <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">H</font> &amp; <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="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Q</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:29_2_1"/>
 <a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c4">G</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T48" title="TMAP_1:th.48">TMAP_1:48</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:29_2"/>
 <a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c4">G</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49" title="TMAP_1:th.49">TMAP_1:49</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:29"/>
(  <a href="urysohn3.html#K6" title="URYSOHN3:func.6">Thunder</a> <font color="Maroon" title="c4">G</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds <br/> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; ( <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> implies <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> implies <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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) ) ) )
 <b>by </b><i><a class="txt" href="urysohn3.html#E11:29"><i><font color="Green" title="E35">A8</font></i></a></i>;<br/>


</div>
