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<div class="add">

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="compts_1.html#V5">being_T4</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">B</font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:14"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="measure5.html#K6">{}</a>  &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 ;<br/>

<b>let </b><font color="Maroon">G</font> be    <a href="urysohn3.html#M2" target="_self">Rain</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E2:14"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K2">DYADIC</a> 
 ;<br/>

<b>set </b><font color="Maroon">s</font> =  <a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>;<br/>
<a NAME="E3:14"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E4:14"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:14_1"/>
(  <a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp;  <a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:14_1"/><b>then </b><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">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" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:14_1"/>
( <font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> is    <a href="urysohn3.html#M1" target="_self">Drizzle</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>,<span class="p1">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:14_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:14_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:14"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E6:14"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="urysohn3.html#K2" target="_self">inf_number_dyadic</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>


</div>
