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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="compts_1.html#V5">being_T4</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">B</font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:32"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a>  &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 ;<br/>

<b>let </b><font color="Maroon">G</font> be    <a href="urysohn3.html#M2" target="_self">Rain</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>

<a NAME="E2:32"/>
1 is   <a href="supinf_1.html#NM3">R_eal</a>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">e1</font> = 1 as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E4:32"/><i><font color="Green">E33</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> 0
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="urysohn1.html#T6">URYSOHN1:6</a></i>;<br/>
<a NAME="E5:32"/><b>then </b><i><font color="Green">E34</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a> 
 
<b>by </b><i><a class="ref" href="urysohn1.html#D4">URYSOHN1:def 4</a></i>;<br/>
<a NAME="E6:32"/><b>then </b>
1 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12">halfline</a> 0</span>)</span> <a href="subset_1.html#K4">\/</a> <a href="urysohn1.html#K2">DYADIC</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:32"/><b>then </b><i><font color="Green">E36</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a> 
 
<b>by </b><i><a class="ref" href="urysohn1.html#D5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E8:32"/><i><font color="Green">E38</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> 0
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="urysohn1.html#T6">URYSOHN1:6</a></i>;<br/>
<a NAME="E9:32"/><b>then </b><i><font color="Green">E41</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a> 
 
<b>by </b><i><a class="ref" href="urysohn1.html#D4">URYSOHN1:def 4</a></i>;<br/>
<a NAME="E10:32"/><b>then </b>
0 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12">halfline</a> 0</span>)</span> <a href="subset_1.html#K4">\/</a> <a href="urysohn1.html#K2">DYADIC</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E11:32"/><b>then </b><i><font color="Green">E42</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a> 
 
<b>by </b><i><a class="ref" href="urysohn1.html#D5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E12:32"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds <br/> ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Olive">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> 1 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1_1"/>
(  <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Maroon">x</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a>  or  <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Maroon">x</font>,<font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_1"><b>case </b></a><a NAME="E1:32_1_1_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Maroon">x</font>,<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 ;<br/><div class="add"><a NAME="E2:32_1_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><a NAME="E3:32_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1_1_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:32_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i>, , , <a class="txt" href="urysohn3.html#E1:32"><i><font color="Green">E31</font></i></a>, </i>;<br/>
<a NAME="E3:32_1_1_1_1_1"/>
( <font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> 0 is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>,0 &amp; ( for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:32_1_1_1_1_1"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:32_1_1_1_1_1"/><b>then </b>
for <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">s</font> <a href="hidden.html#R1">=</a> 1 holds <br/> not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E8:32"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:32_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="urysohn3.html#E7:32"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="urysohn2.html#T31">URYSOHN2:31</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:32_1_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2"><b>case </b></a><a NAME="E1:32_1_1_1_2"/><i><font color="Green">E65</font></i>: 
 <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Maroon">x</font>,<font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">S</font> =  <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Maroon">x</font>,<font color="Maroon">G</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a>  ;<br/><a NAME="E3:32_1_1_1_2"/>
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E4:32_1_1_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">S</font> &amp;  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">S</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e1</font> )
 <b>by </b><i/>;<br/><a NAME="E5:32_1_1_1_2"/><i><font color="Green">E67</font></i>: 
( <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">S</font> &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><a NAME="E6:32_1_1_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i/>;<br/><a NAME="E7:32_1_1_1_2"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1_1_1_2_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:32_1_1_1_2_1"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> 0
 
<b>by </b><i>, , <a class="txt" href="urysohn3.html#E4:32"><i><font color="Green">E33</font></i></a>, <a class="txt" href="urysohn3.html#E5:32"><i><font color="Green">E34</font></i></a>, </i>;<br/>
<a NAME="E3:32_1_1_1_2_1"/>
( <font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> 0 is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>,0 &amp; ( for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:32_1_1_1_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:32_1_1_1_2_1"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">s</font> being  <a href="supinf_1.html#NM3">R_eal</a>  st  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> not <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Maroon">x</font>,<font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:32_1_1_1_2_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/>

<b>assume </b><a NAME="E2:32_1_1_1_2_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Maroon">x</font>,<font color="Maroon">G</font>
 ;<br/>

<a NAME="E3:32_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  &amp; ( for <font color="Olive">t</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> holds <br/> not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> ) )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E5:32_1_1_1_2_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12">halfline</a> 0</span>)</span> <a href="subset_1.html#K4">\/</a> <a href="urysohn1.html#K2">DYADIC</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:32_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a> 
 
<b>by </b><i><a class="ref" href="urysohn1.html#D5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:32_1_1_1_2_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="urysohn1.html#D4">URYSOHN1:def 4</a></i>;<br/>
<a NAME="E9:32_1_1_1_2_1_1"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i>, , , <a class="ref" href="urysohn3.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> as    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">n</font> <b>by </b><i>, </i>;<br/>
<a NAME="E11:32_1_1_1_2_1_1"/>
0 <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="urysohn1.html#T12">URYSOHN1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">r1</font> = 0, <font color="Maroon">r2</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="urysohn3.html#T1" target="_self">Th1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_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="E1:32_1_1_1_2_1_1_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> )
 </span><b>by </b><i><a class="ref" href="urysohn3.html#T1" target="_self">Th1</a>, <a class="ref" href="urysohn1.html#T5">URYSOHN1:5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_1_1_1_2_1_1_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:32_1_1_1_2_1_1_1_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> 0
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:32_1_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E9:32"><i><font color="Green">E41</font></i></a>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_1_1_1_2_1_1_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:32_1_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E80</font></i>: 
(  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="urysohn3.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> <a href="urysohn3.html#K4" target="_self">.</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> 0 )
 <b>by </b><i>, </i>;<br/><a NAME="E3:32_1_1_1_2_1_1_1_2"/>
<font color="Maroon">D</font> <a href="urysohn3.html#K4" target="_self">.</a> <font color="Maroon">r1</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="urysohn3.html#K4" target="_self">.</a> <font color="Maroon">r1</font></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><a NAME="E4:32_1_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">D</font> <a href="urysohn3.html#K4" target="_self">.</a> <font color="Maroon">r1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> <a href="urysohn3.html#K4" target="_self">.</a> <font color="Maroon">r2</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:32_1_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E6:32_1_1_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="urysohn3.html#E9:32"><i><font color="Green">E41</font></i></a>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:32_1_1_1_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Maroon">x</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E7:32"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E8:32_1_1_1_2_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a> 
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E11:32_1_1_1_2_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="urysohn1.html#D4">URYSOHN1:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_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="E1:32_1_1_1_2_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> )
 </span><b>by </b><i>, <a class="ref" href="urysohn1.html#T5">URYSOHN1:5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_1"><b>suppose </b></a><a NAME="E1:32_1_1_1_2_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_1_1_1_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="urysohn3.html#E9:32"><i><font color="Green">E41</font></i></a>, <a class="txt" href="urysohn3.html#E11:32"><i><font color="Green">E42</font></i></a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_1_2_1_2_2"><b>suppose </b></a><a NAME="E1:32_1_1_1_2_1_2_2"/><i><font color="Green">E84</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/><a NAME="E3:32_1_1_1_2_1_2_2"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="urysohn3.html#T1" target="_self">Th1</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><b>hence </b><a NAME="E4:32_1_1_1_2_1_2_2"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="urysohn3.html#E12:32"><i><font color="Green">E44</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E8:32_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1_1_1_2_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:32_1_1_1_2_2"/><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i>, , , <a class="txt" href="urysohn3.html#E1:32"><i><font color="Green">E31</font></i></a>, </i>;<br/>
<a NAME="E3:32_1_1_1_2_2"/>
( <font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> 0 is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>,0 &amp; ( for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:32_1_1_1_2_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:32_1_1_1_2_2"/><b>then </b>
for <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">s</font> <a href="hidden.html#R1">=</a> 1 holds <br/> not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E9:32"><i><font color="Green">E41</font></i></a>, <a class="txt" href="urysohn3.html#E11:32"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:32_1_1_1_2_2"/><b>then </b><i><font color="Green">E88</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Maroon">x</font>,<font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="urysohn2.html#T31">URYSOHN2:31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">S</font> =  <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Maroon">x</font>,<font color="Maroon">G</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a>  ;<br/>
<a NAME="E8:32_1_1_1_2_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">e1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T76">SUPINF_1:76</a></i>;<br/>
<a NAME="E9:32_1_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:32_1_1_1_2_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E12:32"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:32_1_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:32_1_1_1_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i>, <a class="txt" href="urysohn3.html#E8:32"><i><font color="Green">E38</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="E2:32_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 ) )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:32"/>
 <a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font> is <a href="pre_topc.html#V5">continuous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_2"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds   <a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>

<a NAME="E1:32_2_1"/>
for <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>   st <font color="Olive">Q</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">Q</font> holds <br/> ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">H</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">H</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Olive">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:32_2_1_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">Q</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3">REAL</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E3:32_2_1_1"/><i><font color="Green">E93</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="metric_1.html#K8">RealSpace</a>  &amp; the <a href="pre_topc.html#U1">topology</a> of <a href="topmetr.html#K3">R^1</a>  <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <a href="metric_1.html#K8">RealSpace</a>  )
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E4:32_2_1_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="txt" href="urysohn3.html#E7:32"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="txt" href="urysohn3.html#E7:32"><i><font color="Green">E36</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:32_2_1_1"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="txt" href="urysohn3.html#E7:32"><i><font color="Green">E36</font></i></a>, , <a class="txt" href="urysohn3.html#E8:32"><i><font color="Green">E38</font></i></a>, <a class="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>
<a NAME="E8:32_2_1_1"/>
 ex <font color="Olive">r0</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r0</font> &amp; <font color="Olive">r0</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_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="E1:32_2_1_1_1_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 or 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_2_1_1_1_1_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><div class="add"><b>consider </b><font color="Maroon">r0</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:32_2_1_1_1_1_1"/><i><font color="Green">E139</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r0</font> &amp; <font color="Maroon">r0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T7">XREAL_1:7</a></i>;<br/><b>reconsider </b><font color="Maroon">r0</font> = <font color="Maroon">r0</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>take </b>
<font color="Maroon">r0</font>
;<br/><b>thus </b><a NAME="E5:32_2_1_1_1_1_1"/>
( <font color="Maroon">r0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r0</font> &amp; <font color="Maroon">r0</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, <a class="txt" href="urysohn3.html#E9:32"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_2_1_1_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_2_1_1_1_1_2"/>
 ex <font color="Olive">r0</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r0</font> &amp; <font color="Olive">r0</font> <a href="xxreal_0.html#R1">&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">r0</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:32_2_1_1"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">r0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r0</font> &amp; <font color="Maroon">r0</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<b>consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E12:32_2_1_1"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r0</font> )
 <b>by </b><i>, <a class="ref" href="urysohn2.html#T28">URYSOHN2:28</a></i>;<br/>
<a NAME="E13:32_2_1_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="txt" href="urysohn3.html#E9:32"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E14:32_2_1_1"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <a href="urysohn1.html#K2">DYADIC</a>  <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> 1</span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E9:32"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E15:32_2_1_1"/><i><font color="Green">E144</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> holds <br/><span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:32_2_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font>
 ;<br/>

<a NAME="E2:32_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E3:32_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E11:32"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E17:32_2_1_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E9:32"><i><font color="Green">E41</font></i></a>, <a class="ref" href="urysohn1.html#D4">URYSOHN1:def 4</a></i>;<br/>
<a NAME="E18:32_2_1_1"/>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12">halfline</a> 0</span>)</span> <a href="subset_1.html#K4">\/</a> <a href="urysohn1.html#K2">DYADIC</a> 
 
<b>by </b><i><a class="txt" href="urysohn3.html#E9:32"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E19:32_2_1_1"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a> 
 
<b>by </b><i><a class="ref" href="urysohn1.html#D5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E20:32_2_1_1"/><b>then </b><i><font color="Green">E147</font></i>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font>
 
<b>by </b><i>, <a class="txt" href="urysohn3.html#E9:32"><i><font color="Green">E41</font></i></a>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <font color="Maroon">n</font> as    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>,<font color="Maroon">n</font> <b>by </b><i>, </i>;<br/>
<a NAME="E22:32_2_1_1"/><i><font color="Green">E148</font></i>: 
( 0 <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">n</font> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">n</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E9:32"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="urysohn1.html#T12">URYSOHN1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">r1</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">D</font> <a href="urysohn3.html#K4" target="_self">.</a> <font color="Maroon">r1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E25:32_2_1_1"/>
 ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">H</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">H</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Olive">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_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="E1:32_2_1_1_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:32_2_1_1_3_1_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">H</font>
;<br/><a NAME="E2:32_2_1_1_3_1_1"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_1_3_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_2_1_1_3_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:32_2_1_1_3_1_1_1"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E5:32_2_1_1_3_1_1_1"/><i><font color="Green">E152</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x1</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i>, , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E9:32_2_1_1_3_1_1_1"/><i><font color="Green">E153</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15">TOPMETR:15</a></i>;<br/>
<a NAME="E10:32_2_1_1_3_1_1_1"/><i><font color="Green">E154</font></i>: 
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E11:32_2_1_1_3_1_1_1"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">q</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> 0
 
<b>by </b><i><a class="txt" href="urysohn3.html#E12:32"><i><font color="Green">E44</font></i></a>, , , </i>;<br/>
<a NAME="E12:32_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E156</font></i>: 
( <font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 
<b>by </b><i>, , <a class="ref" href="urysohn3.html#D1" target="_self">Def1</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E13:32_2_1_1_3_1_1_1"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font>
 
;<br/>
<a NAME="E14:32_2_1_1_3_1_1_1"/><b>then </b>
(  <a href="real_1.html#K1">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="urysohn3.html#T5" target="_self">Th5</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E15:32_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E157</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>
<a NAME="E16:32_2_1_1_3_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_2_1_1_3_1_1_1_1"/><i><font color="Green">E158</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_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="E1:32_2_1_1_3_1_1_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font> or <font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:32_2_1_1_3_1_1_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_2_1_1_3_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="urysohn3.html#T5" target="_self">Th5</a>, , <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:32_2_1_1_3_1_1_1_1_1_2"/>
<font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:32_2_1_1_3_1_1_1_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E3:32_2_1_1_3_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E17:32_2_1_1_3_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="urysohn3.html#T3" target="_self">Th3</a>, <a class="ref" href="urysohn3.html#T4" target="_self">Th4</a>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:32_2_1_1_3_1_1"/>
( <font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, , , <a class="ref" href="urysohn3.html#T1" target="_self">Th1</a>, , , <a class="txt" href="urysohn3.html#E1"><i><font color="Green">Lemma24</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:32_2_1_1_3_1_2"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:32_2_1_1_3_1_2"/><i><font color="Green">E160</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">p</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><b>consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:32_2_1_1_3_1_2"/><i><font color="Green">E161</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <a href="urysohn1.html#K2">DYADIC</a>  <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> 1</span>)</span> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <a href="urysohn1.html#K2">DYADIC</a>  <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> 1</span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 <b>by </b><i>, , , <a class="ref" href="urysohn2.html#T35">URYSOHN2:35</a></i>;<br/><b>consider </b><font color="Maroon">r4</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:32_2_1_1_3_1_2"/><i><font color="Green">E163</font></i>: 
( <font color="Maroon">r4</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r4</font> &amp; <font color="Maroon">r4</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="urysohn3.html#D1" target="_self">Def1</a>, <a class="ref" href="urysohn2.html#T28">URYSOHN2:28</a></i>;<br/><b>consider </b><font color="Maroon">r3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:32_2_1_1_3_1_2"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">r3</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a>  &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r3</font> &amp; <font color="Maroon">r3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> )
 <b>by </b><i><a class="ref" href="urysohn3.html#D1" target="_self">Def1</a>, <a class="ref" href="urysohn2.html#T29">URYSOHN2:29</a></i>;<br/><b>reconsider </b><font color="Maroon">A</font> = <span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r3</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="urysohn3.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E11:32_2_1_1_3_1_2"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="ref" href="urysohn3.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E12:32_2_1_1_3_1_2"/>
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  or <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 )
 <b>by </b><i><a class="ref" href="urysohn3.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E13:32_2_1_1_3_1_2"/><b>then </b>
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12">halfline</a> 0</span>)</span> <a href="subset_1.html#K4">\/</a> <a href="urysohn1.html#K2">DYADIC</a>  or <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E14:32_2_1_1_3_1_2"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a> 
 <b>by </b><i><a class="ref" href="urysohn1.html#D5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>reconsider </b><font color="Maroon">C</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/><a NAME="E17:32_2_1_1_3_1_2"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T26">TOPS_1:26</a></i>;<br/><a NAME="E18:32_2_1_1_3_1_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/><a NAME="E19:32_2_1_1_3_1_2"/><b>then </b><i><font color="Green">E168</font></i>: 
<font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/><b>take </b><font color="Maroon">H</font> = <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font>;<br/><a NAME="E20:32_2_1_1_3_1_2"/><i><font color="Green">E169</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r4</font>
 <b>by </b><i><a class="ref" href="urysohn3.html#D1" target="_self">Def1</a>, <a class="ref" href="urysohn3.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E21:32_2_1_1_3_1_2"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">r4</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12">halfline</a> 0</span>)</span> <a href="subset_1.html#K4">\/</a> <a href="urysohn1.html#K2">DYADIC</a> 
 <b>by </b><i><a class="ref" href="urysohn3.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E22:32_2_1_1_3_1_2"/><b>then </b>
<font color="Maroon">r4</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a> 
 <b>by </b><i><a class="ref" href="urysohn1.html#D5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E23:32_2_1_1_3_1_2"/><b>then </b><i><font color="Green">E171</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r4</font>
 <b>by </b><i>, <a class="ref" href="urysohn3.html#T3" target="_self">Th3</a>, <a class="ref" href="urysohn3.html#T5" target="_self">Th5</a>, </i>;<br/><b>reconsider </b><font color="Maroon">r4</font> = <font color="Maroon">r4</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K3">DOM</a>  <b>by </b><i>, <a class="ref" href="urysohn1.html#D5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E25:32_2_1_1_3_1_2"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K4" target="_self">.</a> <font color="Maroon">r4</font>
 <b>by </b><i>, <a class="ref" href="urysohn3.html#T3" target="_self">Th3</a>, , </i>;<br/><a NAME="E26:32_2_1_1_3_1_2"/><b>then </b><i><font color="Green">E172</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/><a NAME="E27:32_2_1_1_3_1_2"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E28:32_2_1_1_3_1_2"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r3</font>
 <b>by </b><i>, <a class="ref" href="urysohn3.html#T4" target="_self">Th4</a>, <a class="txt" href="urysohn3.html#E1"><i><font color="Green">Lemma24</font></i></a></i>;<br/><a NAME="E29:32_2_1_1_3_1_2"/>
<span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_1_3_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:32_2_1_1_3_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:32_2_1_1_3_1_2_1"/><i><font color="Green">E175</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E6:32_2_1_1_3_1_2_1"/><i><font color="Green">E176</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r3</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:32_2_1_1_3_1_2_1"/>
not <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_1_3_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_2_1_1_3_1_2_1_1"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:32_2_1_1_3_1_2_1_1"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<b>hence </b><a NAME="E3:32_2_1_1_3_1_2_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:32_2_1_1_3_1_2_1"/><b>then </b><i><font color="Green">E178</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x1</font>
 
<b>by </b><i>, <a class="ref" href="urysohn3.html#T5" target="_self">Th5</a>, <a class="txt" href="urysohn3.html#E1"><i><font color="Green">Lemma24</font></i></a></i>;<br/>
<a NAME="E9:32_2_1_1_3_1_2_1"/>
( ( <font color="Maroon">r3</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K12">halfline</a> 0</span>)</span> <a href="subset_1.html#K4">\/</a> <a href="urysohn1.html#K2">DYADIC</a>  or <font color="Maroon">r3</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 ) &amp; not <font color="Maroon">r3</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="urysohn3.html#T4" target="_self">Th4</a>, <a class="ref" href="urysohn1.html#D5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E10:32_2_1_1_3_1_2_1"/><b>then </b>
( ( <font color="Maroon">r3</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 or <font color="Maroon">r3</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  or <font color="Maroon">r3</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 ) &amp; not <font color="Maroon">r3</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E11:32_2_1_1_3_1_2_1"/><b>then </b>
( <font color="Maroon">r3</font> <a href="hidden.html#R2">in</a> <a href="urysohn1.html#K2">DYADIC</a>  <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4">right_open_halfline</a> 1</span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r3</font> )
 
<b>by </b><i><a class="ref" href="urysohn1.html#T1">URYSOHN1:1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E12:32_2_1_1_3_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r3</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:32_2_1_1_3_1_2_1"/><b>then </b><i><font color="Green">E179</font></i>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font>
 
<b>by </b><i><a class="ref" href="urysohn3.html#T4" target="_self">Th4</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  ;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i>, , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E18:32_2_1_1_3_1_2_1"/><i><font color="Green">E180</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T15">TOPMETR:15</a></i>;<br/>
<a NAME="E19:32_2_1_1_3_1_2_1"/><i><font color="Green">E181</font></i>: 
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> implies <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E20:32_2_1_1_3_1_2_1"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">q</font> <a href="real_1.html#K5">-</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="urysohn3.html#D1" target="_self">Def1</a>, , , <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E21:32_2_1_1_3_1_2_1"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="urysohn3.html#D1" target="_self">Def1</a>, , , <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<a NAME="E22:32_2_1_1_3_1_2_1"/><b>then </b><i><font color="Green">E184</font></i>: 
( <font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="urysohn3.html#D1" target="_self">Def1</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E23:32_2_1_1_3_1_2_1"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font>
 
;<br/>
<a NAME="E24:32_2_1_1_3_1_2_1"/><b>then </b>
(  <a href="real_1.html#K1">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E25:32_2_1_1_3_1_2_1"/><b>then </b><i><font color="Green">E185</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>
<a NAME="E26:32_2_1_1_3_1_2_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_1_3_1_2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_2_1_1_3_1_2_1_2"/><i><font color="Green">E186</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_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="E1:32_2_1_1_3_1_2_1_2_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font> or <font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1_3_1_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:32_2_1_1_3_1_2_1_2_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_2_1_1_3_1_2_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="urysohn3.html#D1" target="_self">Def1</a>, , , <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_2_1_1_3_1_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:32_2_1_1_3_1_2_1_2_1_2"/>
<font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:32_2_1_1_3_1_2_1_2_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K5">-</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E3:32_2_1_1_3_1_2_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="urysohn3.html#D1" target="_self">Def1</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E27:32_2_1_1_3_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E30:32_2_1_1_3_1_2"/>
( <font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, , , , <a class="ref" href="tops_1.html#T38">TOPS_1:38</a>, <a class="ref" href="xboole_0.html#D3">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="E26:32_2_1_1"/>
 ex <font color="Olive">H</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">H</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">H</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Olive">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:32_2_1"/>
 <a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T48">TMAP_1:48</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:32_2"/>
 <a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49">TMAP_1:49</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:32"/>
(  <a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font> is <a href="pre_topc.html#V5">continuous</a> &amp; ( for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds <br/> ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Olive">x</font> &amp; <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> implies <span class="p1">(<span class="default"><a href="urysohn3.html#K7" target="_self">Thunder</a> <font color="Maroon">G</font></span>)</span> <a href="urysohn3.html#K8" target="_self">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> 1 ) ) ) )
 <b>by </b><i/>;<br/>


</div>
