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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="compts_1.html#V5" title="COMPTS_1:attr.5">being_T4</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="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a>  &amp; <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c4">G</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>let </b><font color="Maroon" title="c5">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c6">C</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>assume </b><b>that </b><br/><a NAME="E2:18"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">r</font>
 <b>and </b><br/><a NAME="E3:18"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K3" title="URYSOHN1:func.3">DOM</a> 
 ;<br/>

<a NAME="E4:18"/><i><font color="Green" title="E20">A4</font></i>: 
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  or <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E3:18"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="urysohn1.html#D5" title="URYSOHN1:def.5">URYSOHN1:def 5</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1"/>
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a>  or <font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1 )
 </span><b>by </b><i><a class="txt" href="urysohn3.html#E4:18"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><b>suppose </b></a><a NAME="E1:18_1_1"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="prob_1.html#K11" title="PROB_1:func.11">halfline</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:18_1_1"/><b>then </b>
<font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure5.html#K6" title="MEASURE5:func.6">{}</a> 
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:18"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="urysohn3.html#E2:18"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="urysohn3.html#E3:18"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></i>;<br/><a NAME="E3:18_1_1"/><b>then </b>
<font color="Maroon" title="c6">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T5" title="PRE_TOPC:th.5">PRE_TOPC:5</a></i>;<br/><b>hence </b><a NAME="E4:18_1_1"/>
<font color="Maroon" title="c6">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2"><b>suppose </b></a><a NAME="E1:18_1_2"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:18_1_2"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="ref" href="urysohn1.html#D4" title="URYSOHN1:def.4">URYSOHN1:def 4</a></i>;<br/><a NAME="E4:18_1_2"/><i><font color="Green" title="E23">A7</font></i>: 
<span class="p1">(<span class="default"><a href="urysohn3.html#K3" title="URYSOHN3:func.3">Tempest</a> <font color="Maroon" title="c4">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:18"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="urysohn3.html#E3:18"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="urysohn3.html#E1:18_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="urysohn3.html#E3:18_1_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">D</font> = <font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c7">n</font> as    <a href="urysohn3.html#M1" title="URYSOHN3:mode.1">Drizzle</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c7">n</font> <b>by </b><i><a class="txt" href="urysohn3.html#E1:18"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="urysohn3.html#D2" target="_self" title="URYSOHN3:def.2">Def2</a></i>;<br/><a NAME="E6:18_1_2"/><i><font color="Green" title="E24">A8</font></i>: 
for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c7">n</font>  st <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r2</font> holds <br/>( <font color="Maroon" title="c8">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Olive" title="b1">r1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Olive" title="b2">r2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Olive" title="b1">r1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:18"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_2_1_1"/>
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font> )
 </span><b>by </b><i><a class="txt" href="urysohn3.html#E3:18_1_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="urysohn1.html#T5" title="URYSOHN1:th.5">URYSOHN1:5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_1_1"><b>case </b></a><a NAME="E1:18_1_2_1_1_1"/><i><font color="Green" title="E25">A9</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:18_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E26">A10</font></i>: 
( 1 is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c5">r</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c7">n</font> &amp; <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 <b>by </b><i><a class="ref" href="urysohn1.html#T12" title="URYSOHN1:th.12">URYSOHN1:12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c5">r</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c7">n</font> <b>by </b><i><a class="txt" href="urysohn3.html#E1:18_1_2_1_1_1"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="urysohn1.html#T12" title="URYSOHN1:th.12">URYSOHN1:12</a></i>;<br/><a NAME="E4:18_1_2_1_1_1"/>
<font color="Maroon" title="c8">D</font> <a href="urysohn3.html#K4" title="URYSOHN3:func.4">.</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:18"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="urysohn3.html#E2:18"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="urysohn3.html#E3:18"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="urysohn3.html#E1:18_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E5:18_1_2_1_1_1"/>
<font color="Maroon" title="c6">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:18"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="urysohn3.html#E2:18_1_2_1_1_1"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_1_2"><b>case </b></a><a NAME="E1:18_1_2_1_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font>
 ;<br/><div class="add"><a NAME="E2:18_1_2_1_1_2"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <font color="Maroon" title="c7">n</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">r</font> )
 <b>by </b><i><a class="ref" href="urysohn1.html#T12" title="URYSOHN1:th.12">URYSOHN1:12</a></i>;<br/><b>hence </b><a NAME="E3:18_1_2_1_1_2"/>
<font color="Maroon" title="c6">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:18"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="urysohn3.html#E3:18_1_2"><i><font color="Green" title="E22">A6</font></i></a>, <a class="txt" href="urysohn3.html#E4:18_1_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="urysohn3.html#E6:18_1_2"><i><font color="Green" title="E24">A8</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="E8:18_1_2"/>
<font color="Maroon" title="c6">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_3"><b>suppose </b></a><a NAME="E1:18_1_3"/>
<font color="Maroon" title="c5">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> 1
 ;<br/><div class="add"><a NAME="E2:18_1_3"/><b>then </b><i><font color="Green" title="E21">A11</font></i>: 
<font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:18"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="urysohn3.html#E2:18"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="urysohn3.html#E3:18"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="urysohn3.html#D4" target="_self" title="URYSOHN3:def.4">Def4</a></i>;<br/><a NAME="E3:18_1_3"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D1" title="PRE_TOPC:def.1">PRE_TOPC:def 1</a></i>;<br/><b>hence </b><a NAME="E4:18_1_3"/>
<font color="Maroon" title="c6">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="urysohn3.html#E2:18_1_3"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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