<?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="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="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</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> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be  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/>









<b>assume </b><a NAME="E1:28"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:28"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="urysohn1.html#K2">DYADIC</a> 
 
<b>by </b><i><a class="ref" href="urysohn3.html#T15" target="_self">Th15</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E3:28"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E4:28"/>
( 0 is   <a href="supinf_1.html#NM3">R_eal</a> &amp; 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">c<sub>8</sub></font> = 0, <font color="Maroon">c<sub>9</sub></font> = 1 as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E6:28"/>
 <a href="urysohn1.html#K2">DYADIC</a>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="urysohn2.html#T32">URYSOHN2:32</a></i>;<br/>
<a NAME="E7:28"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="measure5.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E2:28"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:28"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="urysohn3.html#E3:28"><i><font color="Green">E25</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a></i>;<br/>
<a NAME="E9:28"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is    <a href="supinf_1.html#M1">majorant</a> of <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D9">SUPINF_1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:28"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="txt" href="urysohn3.html#E11:28"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E13:28"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E7:28"><i><font color="Green">E26</font></i></a>, <a class="txt" href="urysohn3.html#E11:28"><i><font color="Green">E28</font></i></a>, <a class="ref" href="measure5.html#D1">MEASURE5:def 1</a>, <a class="ref" href="supinf_1.html#T76">SUPINF_1:76</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E14:28"/>
(  <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">c<sub>6</sub></font> &amp;  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="urysohn3.html#E9:28"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="supinf_1.html#D16">SUPINF_1:def 16</a></i>;<br/>


</div>
