<?xml version="1.0"?>
<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="E1:3"/><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>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> ex <font color="Olive">G</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">a<sub>1</sub></font>, <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> &amp; ( for <font color="Olive">r1</font>, <font color="Olive">r2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">a<sub>1</sub></font>  st <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> holds <br/>( <font color="Olive">G</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">G</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Olive">G</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r2</font> ) ) );<br/>
<a NAME="E2:3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:3"/><i><font color="Green">E34</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="3_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>given </b><font color="Maroon">G</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">n</font>, <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><a NAME="E1:3_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> &amp; ( for <font color="Olive">r1</font>, <font color="Olive">r2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">n</font>  st <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> holds <br/>( <font color="Maroon">G</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">G</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r2</font> ) ) )
 ;<br/>

<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:3_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> &amp; ( for <font color="Olive">r1</font>, <font color="Olive">r2</font>, <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>  st <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> holds <br/>( <font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r2</font> &amp; ( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="urysohn1.html#K1">dyadic</a> <font color="Maroon">n</font> implies <font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> ) ) ) )
 <b>by </b><i>, , <a class="ref" href="urysohn1.html#T32">URYSOHN1:32</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E4:3_1"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> &amp; ( for <font color="Olive">r1</font>, <font color="Olive">r2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="urysohn1.html#K1">dyadic</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>  st <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r2</font> holds <br/>( <font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r2</font> is <a href="pre_topc.html#V3">open</a> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="urysohn1.html#K4">.</a> <font color="Olive">r2</font> ) ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:3"/>
for <font color="Olive">n</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">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div>
