<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="rcomp_2.html#K1">.[</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="rat_1.html#V1">rational</a> ) </span>}</span> ;<br/>
<a NAME="E1:18_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="rcomp_2.html#K1">.[</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="rat_1.html#V1">rational</a> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_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="E1:18_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="rcomp_2.html#K1">.[</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="rat_1.html#V1">rational</a> ) </span>}</span> 
 ;<br/>

<a NAME="E2:18_1_1_1"/><b>then </b>
 ex <font color="Olive">x</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="rcomp_2.html#K1">.[</a></span> &amp; <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="rat_1.html#V1">rational</a> )
 
;<br/>
<b>hence </b><a NAME="E3:18_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="rcomp_2.html#K1">.[</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="rat_1.html#V1">rational</a> ) </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E3:18_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">B</font> is <a href="topgen_3.html#V1" target="_self">point-filtered</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">U1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">U2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="topgen_3.html#D1" target="_self">TOPGEN_3:def 1</a><br/>

<b>assume </b><a NAME="E1:18_1_1_2"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">U1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">U2</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">U1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">U2</font> )
 ;<br/>

<b>consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">q1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:18_1_1_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">q1</font></span><a href="rcomp_2.html#K1">.[</a></span> &amp; <font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">q1</font> is <a href="rat_1.html#V1">rational</a> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">q2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E5:18_1_1_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">U2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">q2</font></span><a href="rcomp_2.html#K1">.[</a></span> &amp; <font color="Maroon">x2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">q2</font> is <a href="rat_1.html#V1">rational</a> )
 <b>by </b><i/>;<br/>
<a NAME="E6:18_1_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U2</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i/>;<br/>
<a NAME="E8:18_1_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">x'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q2</font> )
 
<b>by </b><i>, , , <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/>
<b>set </b><font color="Maroon">y</font> =  <a href="square_1.html#K2">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>;<br/>
<b>set </b><font color="Maroon">q</font> =  <a href="square_1.html#K1">min</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>;<br/>
<a NAME="E9:18_1_1_2"/>
( <font color="Maroon">x2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">x1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q2</font> )
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:18_1_1_2"/><b>then </b>
(  <a href="square_1.html#K2">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q1</font> &amp;  <a href="square_1.html#K2">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> &amp; <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> &amp;  <a href="square_1.html#K1">min</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q1</font> &amp;  <a href="square_1.html#K1">min</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">q2</font> )
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a>, <a class="ref" href="rcomp_2.html#T2">RCOMP_2:2</a></i>;<br/>
<a NAME="E11:18_1_1_2"/><b>then </b>
(  <a href="square_1.html#K2">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> &amp;  <a href="square_1.html#K1">min</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> is <a href="rat_1.html#V1">rational</a> &amp; <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font></span>)</span></span><a href="rcomp_2.html#K1">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font> &amp; <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font></span>)</span></span><a href="rcomp_2.html#K1">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">U2</font> &amp;  <a href="square_1.html#K2">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font> )
 
<b>by </b><i>, , , <a class="ref" href="topreala.html#T12">TOPREALA:12</a>, <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a>, <a class="ref" href="square_1.html#T50">SQUARE_1:50</a></i>;<br/>
<a NAME="E12:18_1_1_2"/><b>then </b>
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font></span>)</span></span><a href="rcomp_2.html#K1">.[</a></span> &amp; <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font></span>)</span></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span>,<span class="p2">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font></span>)</span></span><a href="rcomp_2.html#K1">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">U2</font> )
 
<b>by </b><i><a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<b>hence </b><a NAME="E13:18_1_1_2"/>
 ex <font color="Olive">U</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Olive">U</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="Olive">U</font> &amp; <font color="Olive">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">U2</font> )
 ;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/><a NAME="E3:18_1_1_3"/>
<font color="Maroon">x'</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x'</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>then consider </b><font color="Maroon">q</font> being  <a href="rat_1.html#V1">rational</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:18_1_1_3"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">x'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x'</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 <b>by </b><i><a class="ref" href="rat_1.html#T22">RAT_1:22</a></i>;<br/><b>take </b><font color="Maroon">U</font> = <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">q</font></span><a href="rcomp_2.html#K1">.[</a></span>;<br/><a NAME="E6:18_1_1_3"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>hence </b><a NAME="E7:18_1_1_3"/>
<font color="Maroon">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E8:18_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font>
 <b>by </b><i>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:18_1_1"/><b>then </b>
<font color="Maroon">B</font> is <a href="abian.html#V2">covering</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:18_1_1"/><b>then </b>
 <a href="pre_topc.html#G1">TopStruct</a>(# <a href="numbers.html#K1">REAL</a> ,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">B</font></span>)</span> #) is  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>
 
<b>by </b><i>, , <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:18_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="pre_topc.html#NM1">TopSpace</a> st <br/>( the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp;  ex <font color="Olive">B</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="numbers.html#K1">REAL</a>  st <br/>( the <a href="pre_topc.html#U1">topology</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Olive">B</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Olive">x</font>,<font color="Olive">q</font></span><a href="rcomp_2.html#K1">.[</a></span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Olive">q</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> is <a href="rat_1.html#V1">rational</a> ) </span>}</span>  ) )
 ;<br/>


</div>
