<?xml version="1.0"?>
<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>let </b><font color="Maroon">R</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:88"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">R</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">R</font> is <a href="pre_topc.html#V3">open</a> )
 <b>and </b><br/><a NAME="E2:88"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> : ( <font color="Olive">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>  holds <br/>(  not <font color="Olive">f</font> is <a href="pre_topc.html#V5">continuous</a> or  not  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> or  not <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or  not <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">q</font> ) ) ) </span>}</span> 
 ;<br/>

<a NAME="E3:88"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:88_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E3:88_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>  holds <br/>(  not <font color="Olive">f</font> is <a href="pre_topc.html#V5">continuous</a> or  not  <a href="relset_1.html#K5">rng</a> <font color="Olive">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> or  not <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or  not <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> ) ) )
 <b>by </b><i><a class="ref" href="jordan2c.html#T12" target="_self">Th12</a></i>;<br/>
<b>thus </b><a NAME="E4:88_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T13" target="_self">Th13</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">P'</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">u</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><b>assume </b><a NAME="E2:88_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><b>reconsider </b><font color="Maroon">R'</font> = <font color="Maroon">R</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span> ;<br/><b>consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:88_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R'</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T11" target="_self">Th11</a>, , <a class="ref" href="jordan2c.html#T13" target="_self">Th13</a>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a></i>;<br/><b>reconsider </b><font color="Maroon">r'</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E7:88_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r'</font>
 <b>by </b><i>, <a class="ref" href="tbsp_1.html#T16">TBSP_1:16</a></i>;<br/><b>take </b><font color="Maroon">r</font> = <font color="Maroon">r</font>;<br/><b>thus </b><a NAME="E8:88_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E9:88_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:88_2_1"/>
not  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P'</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:88_2_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:88_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:88_2_1_1"/>
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or  ex <font color="Olive">f1</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">f1</font> is <a href="pre_topc.html#V5">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">f1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> &amp; <font color="Olive">f1</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Olive">f1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> ) )
 </span><b>by </b><i><a class="ref" href="jordan2c.html#T12" target="_self">Th12</a>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_2_1_1_1"><b>suppose </b></a><a NAME="E1:88_2_1_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:88_2_1_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 ;<br/><a NAME="E2:88_2_1_1_1_1"/>
 ex <font color="Olive">p3</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Olive">p3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Olive">p3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">f1</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>  holds <br/>(  not <font color="Olive">f1</font> is <a href="pre_topc.html#V5">continuous</a> or  not  <a href="relset_1.html#K5">rng</a> <font color="Olive">f1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> or  not <font color="Olive">f1</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or  not <font color="Olive">f1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">p3</font> ) ) )
 <b>by </b><i><a class="ref" href="jordan2c.html#T12" target="_self">Th12</a>, <a class="ref" href="jordan2c.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E3:88_2_1_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E3:88_2_1_1_1"/><b>then </b>
( <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r'</font> )
 <b>by </b><i>, <a class="ref" href="tbsp_1.html#T16">TBSP_1:16</a></i>;<br/><b>then consider </b><font color="Maroon">f2</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E5:88_2_1_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">f2</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r'</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E6:88_2_1_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">f2</font> is <a href="pre_topc.html#V5">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i>, , , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E7:88_2_1_1_1"/>
not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:88_2_1_1_1_2"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:88_2_1_1_1_2"/><b>then </b>
 ex <font color="Olive">q4</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">q4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Olive">q4</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Olive">q4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">f1</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>  holds <br/>(  not <font color="Olive">f1</font> is <a href="pre_topc.html#V5">continuous</a> or  not  <a href="relset_1.html#K5">rng</a> <font color="Olive">f1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> or  not <font color="Olive">f1</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or  not <font color="Olive">f1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">q4</font> ) ) )
 
<b>by </b><i><a class="ref" href="jordan2c.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b><a NAME="E3:88_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:88_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="88_2_1_1_2"><b>suppose </b></a><a NAME="E1:88_2_1_1_2"/><i><font color="Green">E130</font></i>: 
 ex <font color="Olive">f1</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">f1</font> is <a href="pre_topc.html#V5">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">f1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> &amp; <font color="Olive">f1</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Olive">f1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 ;<br/><div class="add"><a NAME="E2:88_2_1_1_2"/>
not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:88_2_1_1_2_1"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:88_2_1_1_2_1"/><b>then </b>
 ex <font color="Olive">q4</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">q4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> &amp; <font color="Olive">q4</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Olive">q4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; ( for <font color="Olive">f1</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>  holds <br/>(  not <font color="Olive">f1</font> is <a href="pre_topc.html#V5">continuous</a> or  not  <a href="relset_1.html#K5">rng</a> <font color="Olive">f1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> or  not <font color="Olive">f1</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or  not <font color="Olive">f1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">q4</font> ) ) )
 
<b>by </b><i><a class="ref" href="jordan2c.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b><a NAME="E3:88_2_1_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T15" target="_self">Th15</a>, , , </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:88_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan2c.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E10:88_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P'</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:88"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T22">TOPMETR:22</a></i>;<br/>


</div>
