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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:20"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="urysohn3.html#M2" target="_self">Rain</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E2:20"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K3">DOM</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E3:20"/><b>then </b><i><font color="Green">E20</font></i>: 
( ( <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 ) &amp; ( <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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="urysohn1.html#D5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E4:20"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1"/>
( ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 ) or ( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="urysohn3.html#E3:20"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><b>suppose </b></a><a NAME="E1:20_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 )
 ;<br/><div class="add"><a NAME="E2:20_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn3.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="urysohn3.html#E4:20"><i><font color="Green">E21</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E3:20_1_1"/>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T22">TOPS_1:22</a></i>;<br/><a NAME="E4:20_1_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:20_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/><b>hence </b><a NAME="E5:20_1_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2"><b>suppose </b></a><a NAME="E1:20_1_2"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 )
 ;<br/><div class="add"><a NAME="E2:20_1_2"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="urysohn1.html#T1">URYSOHN1:1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:20_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:20_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="urysohn1.html#D4">URYSOHN1:def 4</a></i>;<br/><a NAME="E5:20_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E4:20_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="urysohn1.html#T5">URYSOHN1:5</a></i>;<br/><b>hence </b><a NAME="E6:20_1_2"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="urysohn3.html#E2:20_1_2"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_3"><b>suppose </b></a><a NAME="E1:20_1_3"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 )
 ;<br/><div class="add"><a NAME="E2:20_1_3"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="urysohn1.html#T1">URYSOHN1:1</a></i>;<br/><a NAME="E3:20_1_3"/>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:20_1_3"><i><font color="Green">E22</font></i></a>, <a class="ref" href="urysohn1.html#T2">URYSOHN1:2</a></i>;<br/><a NAME="E4:20_1_3"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>hence </b><a NAME="E5:20_1_3"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="urysohn3.html#E2:20_1_3"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_4"><b>suppose </b></a><a NAME="E1:20_1_4"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  )
 ;<br/><div class="add"><a NAME="E2:20_1_4"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn3.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="urysohn3.html#E4:20"><i><font color="Green">E21</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E3:20_1_4"/>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T22">TOPS_1:22</a></i>;<br/><a NAME="E4:20_1_4"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:20_1_4"><i><font color="Green">E22</font></i></a>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/><b>hence </b><a NAME="E5:20_1_4"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_5"><b>suppose </b></a><a NAME="E1:20_1_5"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:20_1_5"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="urysohn1.html#D4">URYSOHN1:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:20_1_5"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:20_1_5"><i><font color="Green">E22</font></i></a>, <a class="ref" href="urysohn1.html#D4">URYSOHN1:def 4</a></i>;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font>;<br/><a NAME="E6:20_1_5"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:20_1_5"/><b>then </b><i><font color="Green">E25</font></i>: 
(  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp;  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="urysohn2.html#T33">URYSOHN2:33</a></i>;<br/><a NAME="E8:20_1_5"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn3.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="urysohn3.html#E1:20_1_5"><i><font color="Green">E22</font></i></a>, <a class="txt" href="urysohn3.html#E3:20_1_5"><i><font color="Green">E23</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self">Def4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> as    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="urysohn3.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="ref" href="urysohn3.html#D2" target="_self">Def2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <b>by </b><i><a class="txt" href="urysohn3.html#E3:20_1_5"><i><font color="Green">E23</font></i></a>, <a class="txt" href="urysohn3.html#E5:20_1_5"><i><font color="Green">E24</font></i></a>, <a class="txt" href="urysohn3.html#E7:20_1_5"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E11:20_1_5"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="urysohn3.html#K4" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> <a href="urysohn3.html#K4" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn3.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="urysohn3.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E12:20_1_5"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn3.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="urysohn3.html#E4:20"><i><font color="Green">E21</font></i></a>, <a class="txt" href="urysohn3.html#E1:20_1_5"><i><font color="Green">E22</font></i></a>, <a class="txt" href="urysohn3.html#E8:20_1_5"><i><font color="Green">E26</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_6"><b>suppose </b></a><a NAME="E1:20_1_6"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  )
 ;<br/><div class="add"><a NAME="E2:20_1_6"/><b>then </b><i><font color="Green">E23</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="urysohn1.html#T2">URYSOHN1:2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:20_1_6"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:20_1_6"><i><font color="Green">E22</font></i></a>, <a class="ref" href="urysohn1.html#D4">URYSOHN1:def 4</a></i>;<br/><a NAME="E5:20_1_6"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="urysohn3.html#E4:20_1_6"><i><font color="Green">E24</font></i></a>, <a class="ref" href="urysohn1.html#T5">URYSOHN1:5</a></i>;<br/><b>hence </b><a NAME="E6:20_1_6"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="urysohn3.html#E2:20_1_6"><i><font color="Green">E23</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="20_1_7"><b>suppose </b></a><a NAME="E1:20_1_7"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K12">halfline</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 )
 ;<br/><div class="add"><a NAME="E2:20_1_7"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn3.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="urysohn3.html#E4:20"><i><font color="Green">E21</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E3:20_1_7"/>
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T22">TOPS_1:22</a></i>;<br/><a NAME="E4:20_1_7"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:20_1_7"><i><font color="Green">E22</font></i></a>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/><b>hence </b><a NAME="E5:20_1_7"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_8"><b>suppose </b></a><a NAME="E1:20_1_8"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 )
 ;<br/><div class="add"><a NAME="E2:20_1_8"/><b>then </b>
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn3.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E3:20_1_8"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_9"><b>suppose </b></a><a NAME="E1:20_1_9"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#K4">right_open_halfline</a> 1 )
 ;<br/><div class="add"><a NAME="E2:20_1_9"/><b>then </b>
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" target="_self">Tempest</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="urysohn3.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E3:20_1_9"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
