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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="E1:5"/><i><font color="Green" title="E5">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">n</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c5">G</font> be    <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="c4">n</font>;<br/>



<a NAME="E2:5"/><i><font color="Green" title="E6">A2</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &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="c5">G</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#E1:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></i>;<br/>
<a NAME="E3:5"/>
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="c4">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="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1: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="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1: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="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1: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="c5">G</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r2</font> )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>, <a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E5:5"/><i><font color="Green" title="E7">A3</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 &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="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; ( for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">r</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>  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="c6">F</font> <a href="urysohn1.html#K4" title="URYSOHN1: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="c6">F</font> <a href="urysohn1.html#K4" title="URYSOHN1: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="c6">F</font> <a href="urysohn1.html#K4" title="URYSOHN1: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="c6">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r2</font> &amp; ( <font color="Olive" title="b3">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="c4">n</font> implies <font color="Maroon" title="c6">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">r</font> ) ) ) )
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="urysohn3.html#E2:5"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="urysohn1.html#T32" title="URYSOHN1:th.32">URYSOHN1:32</a></i>;<br/>
<a NAME="E6:5"/>
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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>  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="c6">F</font> <a href="urysohn1.html#K4" title="URYSOHN1: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="c6">F</font> <a href="urysohn1.html#K4" title="URYSOHN1: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="c6">F</font> <a href="urysohn1.html#K4" title="URYSOHN1: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="c6">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Olive" title="b2">r2</font> )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E5:5"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c6">F</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="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <b>by </b><i><a class="txt" href="urysohn3.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="urysohn3.html#E5:5"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="urysohn3.html#D1" target="_self" title="URYSOHN3:def.1">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">F</font>
;<br/>

<b>let </b><font color="Maroon" title="c8">r</font> be    <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/>

<b>assume </b><a NAME="E8:5"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c8">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="c4">n</font>
 ;<br/>

<a NAME="E9:5"/>
( 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="urysohn1.html#T12" title="URYSOHN1:th.12">URYSOHN1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">r1</font>, <font color="Maroon" title="c10">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span><b> such that </b><br/><a NAME="E11:5"/><i><font color="Green" title="E9">A5</font></i>: 
( <font color="Maroon" title="c9">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c10">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/>
<b>thus </b><a NAME="E12:5"/>
<font color="Maroon" title="c7">F</font> <a href="urysohn1.html#K4" title="URYSOHN1:func.4">.</a> <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="txt" href="urysohn3.html#E5:5"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="urysohn3.html#E8:5"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="urysohn3.html#E11:5"><i><font color="Green" title="E9">A5</font></i></a></i>;<br/>


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