<?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="tdlat_3.html#V1">discrete</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/>

<b>set </b><font color="Maroon">B0</font> =  <a href="eqrel_1.html#K11">SmallestPartition</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>;<br/>
<a NAME="E1:25"/><i><font color="Green">E29</font></i>: 
 <a href="eqrel_1.html#K11">SmallestPartition</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T46">EQREL_1:46</a></i>;<br/>
<a NAME="E2:25"/><i><font color="Green">E31</font></i>: 
 <a href="eqrel_1.html#K11">SmallestPartition</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K11">SmallestPartition</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:25_1"/>
<font color="Maroon">a</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tdlat_3.html#T17">TDLAT_3:17</a></i>;<br/>
<b>hence </b><a NAME="E4:25_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:25_2"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E2:25_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>reconsider </b><font color="Maroon">a</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/><b>take </b><font color="Maroon">a</font> = <font color="Maroon">a</font>;<br/><b>thus </b><a NAME="E4:25_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K11">SmallestPartition</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:25_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>thus </b><a NAME="E6:25_2"/>
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:25"/><i><font color="Green">E36</font></i>: 
 <a href="eqrel_1.html#K11">SmallestPartition</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T32">YELLOW_9:32</a></i>;<br/>

<b>let </b><font color="Maroon">B</font> be    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>



<b>assume </b><a NAME="E5:25"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K11">SmallestPartition</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E7:25"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E9:25"/>
( <font color="Maroon">a</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="yellow_8.html#T19">YELLOW_8:19</a></i>;<br/>
<b>then consider </b><font color="Maroon">U</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E11:25"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; <font color="Maroon">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="yellow_9.html#T31">YELLOW_9:31</a></i>;<br/>
<a NAME="E12:25"/>
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>hence </b><a NAME="E13:25"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
