<?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="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">S</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">T</font>;<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>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">A</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font> ) </span>}</span> ;<br/>
<a NAME="E1:86"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">A</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</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="E1:86_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">A</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:86_1"/><b>then </b>
 ex <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span> &amp; <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font> )
 
;<br/>
<a NAME="E3:86_1"/><b>then </b>
<font color="Maroon">u</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font>
 
;<br/>
<b>hence </b><a NAME="E4:86_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">A</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">S</font> ;<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">U</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:86_2"/>
<font color="Maroon">U</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><b>then consider </b><font color="Maroon">U0</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:86_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">U0</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U0</font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="tsp_1.html#D1">TSP_1:def 1</a></i>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E4:86_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font>
 ;<br/><a NAME="E5:86_2"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U0</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="topgen_5.html#T8" target="_self">Th8</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">V0</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E7:86_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">V0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V0</font> &amp; <font color="Maroon">V0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U0</font> )
 <b>by </b><i><a class="ref" href="topgen_5.html#T8" target="_self">Th8</a>, <a class="ref" href="yellow_9.html#T31">YELLOW_9:31</a></i>;<br/><b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V0</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> ;<br/><b>take </b><font color="Maroon">V</font> = <font color="Maroon">V</font>;<br/><a NAME="E9:86_2"/>
<font color="Maroon">V0</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E10:86_2"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E11:86_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><a NAME="E12:86_2"/>
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font>
 <b>by </b><i><a class="ref" href="topgen_5.html#T8" target="_self">Th8</a>, , <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:86"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</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="E1:86_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:86_3"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font> )
 ;<br/>
<a NAME="E4:86_3"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<br/>
<a NAME="E5:86_3"/><b>then </b>
( <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> &amp; <font color="Maroon">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="topgen_5.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E6:86_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="topgen_5.html#T8" target="_self">Th8</a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:86"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">A</font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font></span>)</span></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">S</font> ) </span>}</span>  is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T32">YELLOW_9:32</a></i>;<br/>


</div>
