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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="E1:13"/><i><font color="Green" title="E13">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">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E2:13"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="urysohn1.html#K2" title="URYSOHN1:func.2">DYADIC</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c6">s</font> =  <a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>;<br/>
<a NAME="E3:13"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E4:13"/><i><font color="Green" title="E16">A4</font></i>: 
for <font color="Olive" title="b1">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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">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>assume </b><a NAME="E1:13_1"/><i><font color="Green" title="E17">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 ;<br/>

<a NAME="E2:13_1"/>
(  <a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n</font> &amp;  <a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E3:13_1"/><b>then </b><i><font color="Green" title="E18">A6</font></i>: 
( <font color="Maroon" title="c5">x</font> <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"><span class="p2">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n</font></span>)</span> &amp; <font color="Maroon" title="c5">x</font> <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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="urysohn3.html#T7" target="_self" title="URYSOHN3:th.7">Th7</a></i>;<br/>
<a NAME="E4:13_1"/>
( <font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n</font></span>)</span> is    <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>,<span class="p1">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n</font> &amp; ( for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> ) )
 
<b>by </b><i><a class="txt" href="urysohn3.html#E1:13"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="urysohn3.html#D2" target="_self" title="URYSOHN3:def.2">Def2</a></i>;<br/>
<a NAME="E5:13_1"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="urysohn1.html#K1" title="URYSOHN1:func.1">dyadic</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:13_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="urysohn3.html#E1:13"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="urysohn3.html#E1:13_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="urysohn3.html#E3:13_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="urysohn3.html#D2" target="_self" title="URYSOHN3:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:13"/>
for <font color="Olive" title="b1">i</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="urysohn3.html#E3:13"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="urysohn3.html#E4:13"><i><font color="Green" title="E16">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:13"/>
for <font color="Olive" title="b1">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>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="urysohn3.html#K2" title="URYSOHN3:func.2">inf_number_dyadic</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 ;<br/>


</div>
