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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c4">R</font> be    <a href="urysohn3.html#M2" title="URYSOHN3:mode.2">Rain</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ] <b>means </b>( (  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> $1,<font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) &amp; ( for <font color="Olive" title="b1">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>   st <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> $1,<font color="Maroon" title="c4">R</font> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Olive" title="b1">S</font> ) );<br/>
<a NAME="E1:24"/><i><font color="Green" title="E22">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>  ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1"/>
(  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  or  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1"/><i><font color="Green" title="E23">A2</font></i>: 
 <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c6">y</font> = 0 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><a NAME="E3:24_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>]
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:24_1_1_1"><i><font color="Green" title="E23">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E4:24_1_1_1"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_2"/>
 <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">R</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">S</font> =  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">R</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>  ;<br/><a NAME="E3:24_1_1_2"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c6">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 <b>by </b><i><a class="ref" href="urysohn3.html#T15" target="_self" title="URYSOHN3:th.15">Th15</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">y</font> =  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c6">S</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> ;<br/><b>reconsider </b><font color="Maroon" title="c8">e1</font> = 1 as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1" title="XXREAL_0:def.1">XXREAL_0:def 1</a></i>;<br/><a NAME="E6:24_1_1_2"/>
 <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> ,<font color="Maroon" title="c8">e1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a>, <a class="ref" href="urysohn2.html#T32" title="URYSOHN2:th.32">URYSOHN2:32</a></i>;<br/><a NAME="E7:24_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="measure5.html#K1" title="MEASURE5:func.1">[.</a><span class="default"><a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> ,<font color="Maroon" title="c8">e1</font></span><a href="measure5.html#K1" title="MEASURE5:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="urysohn3.html#E3:24_1_1_2"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:24_1_1_2"/><b>then </b><i><font color="Green" title="E24">A4</font></i>: 
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c6">S</font> &amp;  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c6">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">e1</font> )
 <b>by </b><i><a class="ref" href="urysohn2.html#T30" title="URYSOHN2:th.30">URYSOHN2:30</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:24_1_1_2"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c9">q</font> as   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a> <b>by </b><i><a class="txt" href="urysohn3.html#E10:24_1_1_2"><i><font color="Green" title="E25">A5</font></i></a></i>;<br/><a NAME="E12:24_1_1_2"/>
(  <a href="supinf_2.html#K7" title="SUPINF_2:func.7">inf</a> <font color="Maroon" title="c6">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c10">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Maroon" title="c6">S</font> )
 <b>by </b><i><a class="txt" href="urysohn3.html#E10:24_1_1_2"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="supinf_1.html#T76" title="SUPINF_1:th.76">SUPINF_1:76</a>, <a class="ref" href="supinf_1.html#T79" title="SUPINF_1:th.79">SUPINF_1:79</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="txt" href="urysohn3.html#E8:24_1_1_2"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="measure6.html#T16" title="MEASURE6:th.16">MEASURE6:16</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/><b>take </b>
<font color="Maroon" title="c11">y</font>
;<br/><b>thus </b><a NAME="E14:24_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c11">y</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:24"/>
 ex <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  st <br/>for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font>]
 
<b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="urysohn3.html#E1:24"><i><font color="Green" title="E22">A1</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon" title="c5">F</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E4:24"/><i><font color="Green" title="E23">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 ;<br/>
<b>take </b>
<font color="Maroon" title="c5">F</font>
;<br/>

<b>thus </b><a NAME="E5:24"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( (  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  implies <font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) &amp; ( for <font color="Olive" title="b2">S</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">ExtREAL</a>   st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="urysohn3.html#K5" title="URYSOHN3:func.5">Rainbow</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">R</font> holds <br/><font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K8" title="SUPINF_2:func.8">sup</a> <font color="Olive" title="b2">S</font> ) )
 <b>by </b><i><a class="txt" href="urysohn3.html#E4:24"><i><font color="Green" title="E23">A6</font></i></a></i>;<br/>


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