<?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="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">R</font> be    <a href="urysohn3.html#M2" target="_self">Rain</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>,  <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K3">R^1</a> ] <b>means </b>( (  <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a>  implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( for <font color="Olive">S</font> being 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>   st <font color="Olive">S</font> <a href="hidden.html#R1">=</a>  <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">R</font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <font color="Olive">S</font> ) );<br/>
<a NAME="E1:25"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K3">R^1</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1"/>
(  <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Maroon">x</font>,<font color="Maroon">R</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">R</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="25_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Maroon">x</font>,<font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">y</font> = 0 as   <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E3:25_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:25_1_1_1"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K3">R^1</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_2"/>
 <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Maroon">x</font>,<font color="Maroon">R</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">R</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="E3:25_1_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="urysohn1.html#K2">DYADIC</a> 
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">y</font> =  <a href="supinf_1.html#K6">sup</a> <font color="Maroon">S</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/><a NAME="E5:25_1_1_2"/>
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="E7:25_1_1_2"/>
 <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"><a href="supinf_2.html#K1">0.</a> ,<font color="Maroon">e1</font></span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a>, <a class="ref" href="urysohn2.html#T32">URYSOHN2:32</a></i>;<br/><a NAME="E8:25_1_1_2"/><b>then </b>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="measure5.html#K1">[.</a><span class="default"><a href="supinf_2.html#K1">0.</a> ,<font color="Maroon">e1</font></span><a href="measure5.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E9:25_1_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7">inf</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><a class="ref" href="urysohn2.html#T30">URYSOHN2:30</a></i>;<br/><b>consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:25_1_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</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">q</font> = <font color="Maroon">q</font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i/>;<br/><a NAME="E13:25_1_1_2"/>
(  <a href="supinf_1.html#K7">inf</a> <font color="Maroon">S</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</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>, <a class="ref" href="supinf_1.html#T79">SUPINF_1:79</a></i>;<br/><a NAME="E14:25_1_1_2"/><b>then </b>
 <a href="supinf_1.html#K7">inf</a> <font color="Maroon">S</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="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E15:25_1_1_2"/><b>then </b>
(  <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><a class="txt" href="urysohn3.html#E1"><i><font color="Green">Lemma24</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="measure6.html#T16">MEASURE6:16</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><b>take </b>
<font color="Maroon">y</font>
;<br/><b>thus </b><a NAME="E17:25_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:25"/>
 ex <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  st <br/>for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">F</font> <a href="urysohn3.html#K6" target="_self">.</a> <font color="Olive">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>then consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<a href="topmetr.html#K3">R^1</a> <b> such that </b><br/><a NAME="E4:25"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="urysohn3.html#K6" target="_self">.</a> <font color="Olive">x</font>]
 ;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E5:25"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds <br/> ( (  <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Olive">p</font>,<font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="measure5.html#K6">{}</a>  implies <font color="Maroon">F</font> <a href="urysohn3.html#K6" target="_self">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( for <font color="Olive">S</font> being 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>   st <font color="Olive">S</font> <a href="hidden.html#R1">=</a>  <a href="urysohn3.html#K5" target="_self">Rainbow</a> <font color="Olive">p</font>,<font color="Maroon">R</font> holds <br/><font color="Maroon">F</font> <a href="urysohn3.html#K6" target="_self">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K6">sup</a> <font color="Olive">S</font> ) )
 <b>by </b><i/>;<br/>


</div>
