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

<b>let </b><font color="Maroon">f</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<a href="borsuk_1.html#K20">I[01]</a> ;<br/>

<b>assume </b><a NAME="E1:25"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a>   holds <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font>
 ;<br/>

<a NAME="E2:25"/><i><font color="Green">E39</font></i>: 
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1 <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#D8">TOPMETR:def 8</a></i>;<br/>
<a NAME="E3:25"/><i><font color="Green">E43</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>,<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<a NAME="E5:25"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <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="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Olive">a</font></span><a href="rcomp_1.html#K1">.]</a></span> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">b</font>,1</span><a href="rcomp_1.html#K1">.]</a></span> ) </span>}</span> ;<br/>
<a NAME="E6:25"/>
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Olive">a</font></span><a href="rcomp_1.html#K1">.]</a></span> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<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">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="E1:25_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Olive">a</font></span><a href="rcomp_1.html#K1">.]</a></span> ) </span>}</span> 
 ;<br/>

<a NAME="E2:25_1"/><b>then </b>
 ex <font color="Olive">a</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Olive">a</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 
;<br/>
<b>hence </b><a NAME="E3:25_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Olive">a</font></span><a href="rcomp_1.html#K1">.]</a></span> ) </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E8:25"/>
<span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">b</font>,1</span><a href="rcomp_1.html#K1">.]</a></span> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><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="E1:25_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b</font> where <font color="Olive">b</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">b</font>,1</span><a href="rcomp_1.html#K1">.]</a></span> ) </span>}</span> 
 ;<br/>

<a NAME="E2:25_2"/><b>then </b>
 ex <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">b</font>,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 
;<br/>
<b>hence </b><a NAME="E3:25_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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"> <font color="Olive">b</font> where <font color="Olive">b</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">b</font>,1</span><a href="rcomp_1.html#K1">.]</a></span> ) </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E10:25"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><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="E1:25_3"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:25_3"/>
 ex <font color="Olive">a0</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">a0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">a0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Olive">a0</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i><a class="ref" href="treal_1.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E4:25_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:25"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4"><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="E1:25_4"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:25_4"/>
 ex <font color="Olive">b0</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">b0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">b0</font>,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i><a class="txt" href="treal_1.html#E4"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:25_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:25"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E13:25"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_5"><b>proof </b></a><div class="add">

<a NAME="E1:25_5"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="treal_1.html#T2" target="_self">Th2</a>, <a class="ref" href="treal_1.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E2:25_5"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_5_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="E1:25_5_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:25_5_1"/>
 ex <font color="Olive">p</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="treal_1.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E4:25_5_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Maroon">y</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">y</font>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:25_5_1"/>
( <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i><a class="txt" href="treal_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:25_5_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_5_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_5_1_1_1"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Maroon">y</font></span><a href="rcomp_1.html#K1">.]</a></span> or <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">y</font>,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 </span><b>by </b><i><a class="txt" href="treal_1.html#E1:25"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_5_1_1_1_1"><b>suppose </b></a><a NAME="E1:25_5_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Maroon">y</font></span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><div class="add"><a NAME="E2:25_5_1_1_1_1"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E3:25_5_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_5_1_1_1_2"><b>suppose </b></a><a NAME="E1:25_5_1_1_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">y</font>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/><div class="add"><a NAME="E2:25_5_1_1_1_2"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E3:25_5_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:25_5_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:25_5"/>
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="txt" href="treal_1.html#E5"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:25"/><i><font color="Green">E73</font></i>: 
1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_6"><b>proof </b></a><div class="add">

<a NAME="E1:25_6"/>
1 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<a NAME="E2:25_6"/><b>then </b><i><font color="Green">E76</font></i>: 
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E3:25_6"/>
( <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i><a class="txt" href="treal_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:25_6"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:25_6"/>
1 <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E15:25"/><i><font color="Green">E77</font></i>: 
0 <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_7"><b>proof </b></a><div class="add">

<a NAME="E1:25_7"/>
0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<a NAME="E2:25_7"/><b>then </b><i><font color="Green">E91</font></i>: 
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E3:25_7"/>
( <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i><a class="txt" href="treal_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:25_7"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="treal_1.html#E1:25"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:25_7"/>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="treal_1.html#E1:25"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:25"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_8"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>;<br/>
<a NAME="E3:25_8"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:25_8"/><b>then </b><i><font color="Green">E114</font></i>: 
 ex <font color="Olive">z</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Olive">z</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 
;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E6:25_8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:25_8"/><b>then </b>
 ex <font color="Olive">b0</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">b0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">b0</font>,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 
;<br/>
<a NAME="E8:25_8"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Maroon">x</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x</font>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="treal_1.html#E6"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:25_8"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="treal_1.html#E6"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E10:25_8"/><b>then </b>
 ex <font color="Olive">u</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
;<br/>
<a NAME="E11:25_8"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="treal_1.html#T4" target="_self">Th4</a>, <a class="ref" href="topmetr2.html#T1">TOPMETR2:1</a></i>;<br/>
<a NAME="E12:25_8"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E13:25_8"/>
contradiction
 <b>by </b><i>, <a class="txt" href="treal_1.html#E6"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:25"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E18:25"/>
 ex <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_1.html#K20">I[01]</a>  st <br/>(  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a>  <a href="hidden.html#R1">=</a> <font color="Olive">P</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">Q</font> &amp; <font color="Olive">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <a href="borsuk_1.html#K20">I[01]</a>  &amp; <font color="Olive">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <a href="borsuk_1.html#K20">I[01]</a>  &amp; <font color="Olive">P</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">Q</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_9"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">A</font>, <font color="Maroon">Q</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_1.html#K20">I[01]</a>  <b>by </b><i><a class="ref" href="treal_1.html#T2" target="_self">Th2</a>, <a class="ref" href="treal_1.html#T3" target="_self">Th3</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<b>take </b>
<font color="Maroon">Q</font>
;<br/>

<b>thus </b><a NAME="E2:25_9"/><i><font color="Green">E117</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>

<b>thus </b><a NAME="E3:25_9"/>
( <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <a href="borsuk_1.html#K20">I[01]</a>  &amp; <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <a href="borsuk_1.html#K20">I[01]</a>  )
 <b>by </b><i><a class="txt" href="treal_1.html#E5"><i><font color="Green">Lemma30</font></i></a>, </i>;<br/>

<b>thus </b><a NAME="E4:25_9"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_9_1"/>
not <font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E2:25_9_1"/><b>then </b><i><font color="Green">E119</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E3:25_9_1"/><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_9_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_9_1_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:25_9_1_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E3:25_9_1_1"/><b>then </b><i><font color="Green">E121</font></i>: 
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E4:25_9_1_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#T25">PRE_TOPC:25</a></i>;<br/>
<b>hence </b><a NAME="E5:25_9_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="treal_1.html#E6"><i><font color="Green">Lemma35</font></i></a>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>;<br/>
<a NAME="E5:25_9_1"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 
<b>by </b><i><a class="ref" href="treal_1.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a>  ;<br/>
<b>reconsider </b><font color="Maroon">t0</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E8:25_9_1"/><i><font color="Green">E124</font></i>: 
 ex <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t0</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Olive">c</font>,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E9:25_9_1"/><b>then </b><i><font color="Green">E125</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t0</font> &amp; <font color="Maroon">t0</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">s0</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t0</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E11:25_9_1"/>
<font color="Maroon">t0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s0</font>
 
<b>by </b><i>, <a class="txt" href="treal_1.html#E2:25"><i><font color="Green">E39</font></i></a>, </i>;<br/>
<a NAME="E12:25_9_1"/><b>then </b><i><font color="Green">E126</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">z</font>, <font color="Maroon">n</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>;<br/>
<b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>;<br/>
<a NAME="E14:25_9_1"/>
<font color="Maroon">s0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">t0</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:25_9_1"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E16:25_9_1"/><b>then </b>
( 0 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E17:25_9_1"/><b>then </b><i><font color="Green">E130</font></i>: 
0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 
;<br/>
<a NAME="E18:25_9_1"/><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> 1
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E19:25_9_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">t0</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>
 
;<br/>
<b>reconsider </b><font color="Maroon">W</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_1.html#K20">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>;<br/>
<a NAME="E21:25_9_1"/><i><font color="Green">E134</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">t0</font>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_9_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> ; <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_9_1_2"/><i><font color="Green">E135</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">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:25_9_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> <b>by </b><i><a class="txt" href="treal_1.html#E10:25"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E5:25_9_1_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <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="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">n</font>,<font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<a NAME="E6:25_9_1_2"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">n</font>,<font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="treal_1.html#E10:25"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E7:25_9_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="heine.html#T1">HEINE:1</a></i>;<br/>
<a NAME="E8:25_9_1_2"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font> &amp; <font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="treal_1.html#E3:25"><i><font color="Green">E43</font></i></a>, <a class="ref" href="absvalue.html#T11">ABSVALUE:11</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:25_9_1_2"/><b>then </b>
<font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:25_9_1_2"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">t0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E11:25_9_1_2"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i>, <a class="txt" href="treal_1.html#E10:25"><i><font color="Green">E46</font></i></a>, </i>;<br/>
<a NAME="E12:25_9_1_2"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">t0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E13:25_9_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">t0</font>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:25_9_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="tbsp_1.html#T16">TBSP_1:16</a></i>;<br/>
<a NAME="E23:25_9_1"/>
( <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">f</font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">z</font> )
 
<b>by </b><i>, <a class="ref" href="tmap_1.html#T55">TMAP_1:55</a>, <a class="ref" href="topmetr.html#T21">TOPMETR:21</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_1.html#K20">I[01]</a> <b> such that </b><br/><a NAME="E25:25_9_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E26:25_9_1"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E27:25_9_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="txt" href="treal_1.html#E10:25"><i><font color="Green">E46</font></i></a>, <a class="ref" href="tmap_1.html#T48">TMAP_1:48</a></i>;<br/>
<b>consider </b><font color="Maroon">s</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="E29:25_9_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E30:25_9_1"/><i><font color="Green">E142</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, , , <a class="ref" href="topmetr.html#T22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</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/>
<b>set </b><font color="Maroon">r0</font> =  <a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>;<br/>
<a NAME="E32:25_9_1"/><i><font color="Green">E143</font></i>: 
 <a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="txt" href="treal_1.html#E11:25"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/>
<a NAME="E33:25_9_1"/><i><font color="Green">E144</font></i>: 
 <a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E34:25_9_1"/><i><font color="Green">E145</font></i>: 
 <a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W0</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_1.html#K20">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>;<br/>
<a NAME="E36:25_9_1"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">W0</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T21">TOPMETR:21</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E37:25_9_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W0</font>
 
<b>by </b><i><a class="ref" href="treal_1.html#T5" target="_self">Th5</a>, <a class="ref" href="tbsp_1.html#T16">TBSP_1:16</a></i>;<br/>
<a NAME="E38:25_9_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="treal_1.html#D2" target="_self">Def2</a>, <a class="ref" href="pcomps_1.html#T1">PCOMPS_1:1</a></i>;<br/>
<a NAME="E39:25_9_1"/><b>then </b><i><font color="Green">E148</font></i>: 
<font color="Maroon">W0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E40:25_9_1"/>
<font color="Maroon">P</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W0</font>
 
<b>by </b><i>, , , <a class="ref" href="pre_topc.html#T54">PRE_TOPC:54</a></i>;<br/>
<a NAME="E41:25_9_1"/><b>then </b><i><font color="Green">E149</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">W0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <a href="borsuk_1.html#K20">I[01]</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">W0</font>;<br/>
<a NAME="E43:25_9_1"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W0</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">w</font> = <font color="Maroon">w</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">w</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">w1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E47:25_9_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <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="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">m</font>,<font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<a NAME="E48:25_9_1"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">m</font>,<font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E49:25_9_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">w</font>,<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="treal_1.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E50:25_9_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> &amp; <font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a>, <a class="ref" href="heine.html#T1">HEINE:1</a></i>;<br/>
<a NAME="E51:25_9_1"/><b>then </b>
<font color="Maroon">d</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E52:25_9_1"/><b>then </b><i><font color="Green">E160</font></i>: 
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E53:25_9_1"/><i><font color="Green">E161</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_9_1_3"><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="E1:25_9_1_3"/><i><font color="Green">E162</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">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:25_9_1_3"/><b>then </b><i><font color="Green">E163</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E5:25_9_1_3"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">t0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="treal_1.html#E2:25"><i><font color="Green">E39</font></i></a>, <a class="txt" href="treal_1.html#E5:25"><i><font color="Green">E45</font></i></a>, , </i>;<br/>
<a NAME="E6:25_9_1_3"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <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="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">n</font>,<font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<a NAME="E7:25_9_1_3"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">n</font>,<font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E8:25_9_1_3"/><b>then </b><i><font color="Green">E165</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="heine.html#T1">HEINE:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_9_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_9_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_9_1_3_1_1"/>
( <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s0</font> or <font color="Maroon">s0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_9_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:25_9_1_3_1_1_1"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s0</font>
 ;<br/><div class="add"><a NAME="E2:25_9_1_3_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E3:25_9_1_3_1_1_1"/><b>then </b>
<font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="treal_1.html#E13:25"><i><font color="Green">E65</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E4:25_9_1_3_1_1_1"/><b>then </b>
<font color="Maroon">s0</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E5:25_9_1_3_1_1_1"/><b>then </b>
<font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>hence </b><a NAME="E6:25_9_1_3_1_1_1"/>
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_9_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:25_9_1_3_1_1_2"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">s0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t</font>
 ;<br/><div class="add"><a NAME="E2:25_9_1_3_1_1_2"/>
<font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">s0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">t0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s0</font>
 <b>by </b><i>, <a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/><a NAME="E3:25_9_1_3_1_1_2"/><b>then </b>
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s0</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:25_9_1_3_1_1_2"/>
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E10:25_9_1_3"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w0</font> where <font color="Olive">w0</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w0</font> &amp; <font color="Olive">w0</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="treal_1.html#E12:25"><i><font color="Green">E64</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:25_9_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E54:25_9_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_9_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:25_9_1_4"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="treal_1.html#D3" target="_self">Def3</a>, , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E2:25_9_1_4"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:25_9_1_4"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">d</font>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E55:25_9_1"/><b>then </b>
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E56:25_9_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:25_9"/>
<font color="Maroon">Q</font> is <a href="pre_topc.html#V4">closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_9_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_9_2"/>
not <font color="Maroon">Q</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E2:25_9_2"/><b>then </b><i><font color="Green">E167</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<a NAME="E3:25_9_2"/><i><font color="Green">E168</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_9_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_9_2_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:25_9_2_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E3:25_9_2_1"/><b>then </b><i><font color="Green">E169</font></i>: 
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E4:25_9_2_1"/>
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#T25">PRE_TOPC:25</a></i>;<br/>
<b>hence </b><a NAME="E5:25_9_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="treal_1.html#E6"><i><font color="Green">Lemma35</font></i></a>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font>;<br/>
<a NAME="E5:25_9_2"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="treal_1.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a>  ;<br/>
<b>reconsider </b><font color="Maroon">t0</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E8:25_9_2"/><i><font color="Green">E171</font></i>: 
 ex <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t0</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Olive">c</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E9:25_9_2"/><b>then </b><i><font color="Green">E172</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t0</font> &amp; <font color="Maroon">t0</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">s0</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t0</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E11:25_9_2"/>
<font color="Maroon">s0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t0</font>
 
<b>by </b><i>, <a class="txt" href="treal_1.html#E2:25"><i><font color="Green">E39</font></i></a>, </i>;<br/>
<a NAME="E12:25_9_2"/><b>then </b><i><font color="Green">E173</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">z</font>, <font color="Maroon">n</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>;<br/>
<b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>;<br/>
<a NAME="E14:25_9_2"/>
<font color="Maroon">t0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s0</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:25_9_2"/><b>then </b><i><font color="Green">E174</font></i>: 
<font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E16:25_9_2"/><b>then </b>
( 0 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K6">/</a> 2 &amp; 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 
<b>by </b><i>, <a class="ref" href="real_1.html#T73">REAL_1:73</a></i>;<br/>
<a NAME="E17:25_9_2"/><b>then </b><i><font color="Green">E175</font></i>: 
0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 
;<br/>
<a NAME="E18:25_9_2"/><i><font color="Green">E176</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> 1
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E19:25_9_2"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">s0</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>
 
;<br/>
<b>reconsider </b><font color="Maroon">W</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_1.html#K20">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>;<br/>
<a NAME="E21:25_9_2"/><i><font color="Green">E178</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Maroon">t0</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_9_2_2"><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="E1:25_9_2_2"/><i><font color="Green">E179</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">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:25_9_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> <b>by </b><i><a class="txt" href="treal_1.html#E10:25"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E5:25_9_2_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <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="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">n</font>,<font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<a NAME="E6:25_9_2_2"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">n</font>,<font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="treal_1.html#E10:25"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E7:25_9_2_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="heine.html#T1">HEINE:1</a></i>;<br/>
<a NAME="E8:25_9_2_2"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font> &amp; <font color="Maroon">t</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="treal_1.html#E3:25"><i><font color="Green">E43</font></i></a>, <a class="ref" href="absvalue.html#T11">ABSVALUE:11</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:25_9_2_2"/><b>then </b>
<font color="Maroon">t</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:25_9_2_2"/><b>then </b><i><font color="Green">E180</font></i>: 
<font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t0</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E11:25_9_2_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i>, <a class="txt" href="treal_1.html#E10:25"><i><font color="Green">E46</font></i></a>, </i>;<br/>
<a NAME="E12:25_9_2_2"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t0</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E13:25_9_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Maroon">t0</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:25_9_2"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="tbsp_1.html#T16">TBSP_1:16</a></i>;<br/>
<a NAME="E23:25_9_2"/>
( <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">f</font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">z</font> )
 
<b>by </b><i>, <a class="ref" href="tmap_1.html#T55">TMAP_1:55</a>, <a class="ref" href="topmetr.html#T21">TOPMETR:21</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_1.html#K20">I[01]</a> <b> such that </b><br/><a NAME="E25:25_9_2"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E26:25_9_2"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E27:25_9_2"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="txt" href="treal_1.html#E10:25"><i><font color="Green">E46</font></i></a>, <a class="ref" href="tmap_1.html#T48">TMAP_1:48</a></i>;<br/>
<b>consider </b><font color="Maroon">s</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="E29:25_9_2"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E30:25_9_2"/><i><font color="Green">E186</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, , , <a class="ref" href="topmetr.html#T22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</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/>
<b>set </b><font color="Maroon">r0</font> =  <a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>;<br/>
<a NAME="E32:25_9_2"/><i><font color="Green">E187</font></i>: 
 <a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="txt" href="treal_1.html#E11:25"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/>
<a NAME="E33:25_9_2"/><i><font color="Green">E188</font></i>: 
 <a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E34:25_9_2"/><i><font color="Green">E189</font></i>: 
 <a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W0</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_1.html#K20">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>;<br/>
<a NAME="E36:25_9_2"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">W0</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T21">TOPMETR:21</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E37:25_9_2"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W0</font>
 
<b>by </b><i><a class="ref" href="treal_1.html#T5" target="_self">Th5</a>, <a class="ref" href="tbsp_1.html#T16">TBSP_1:16</a></i>;<br/>
<a NAME="E38:25_9_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="treal_1.html#D2" target="_self">Def2</a>, <a class="ref" href="pcomps_1.html#T1">PCOMPS_1:1</a></i>;<br/>
<a NAME="E39:25_9_2"/><b>then </b><i><font color="Green">E192</font></i>: 
<font color="Maroon">W0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E40:25_9_2"/>
<font color="Maroon">Q</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W0</font>
 
<b>by </b><i>, , , <a class="ref" href="pre_topc.html#T54">PRE_TOPC:54</a></i>;<br/>
<a NAME="E41:25_9_2"/><b>then </b><i><font color="Green">E193</font></i>: 
<font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">W0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <a href="borsuk_1.html#K20">I[01]</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">W0</font>;<br/>
<a NAME="E43:25_9_2"/><i><font color="Green">E194</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W0</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">w</font> = <font color="Maroon">w</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">w1</font> = <font color="Maroon">w</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">w1</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E47:25_9_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span> <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="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">m</font>,<font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<a NAME="E48:25_9_2"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">m</font>,<font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K1">min</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E49:25_9_2"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">m</font>,<font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="treal_1.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E50:25_9_2"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> &amp; <font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a>, <a class="ref" href="heine.html#T1">HEINE:1</a></i>;<br/>
<a NAME="E51:25_9_2"/><b>then </b>
<font color="Maroon">t0</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">d</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E52:25_9_2"/><b>then </b>
<font color="Maroon">t0</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">d</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E53:25_9_2"/><b>then </b>
<font color="Maroon">t0</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">d</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E54:25_9_2"/><b>then </b><i><font color="Green">E195</font></i>: 
<font color="Maroon">s0</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E55:25_9_2"/><i><font color="Green">E196</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_9_2_3"><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="E1:25_9_2_3"/><i><font color="Green">E197</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">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:25_9_2_3"/><b>then </b><i><font color="Green">E198</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E5:25_9_2_3"/><i><font color="Green">E199</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t0</font> )
 
<b>by </b><i><a class="txt" href="treal_1.html#E2:25"><i><font color="Green">E39</font></i></a>, <a class="txt" href="treal_1.html#E5:25"><i><font color="Green">E45</font></i></a>, , </i>;<br/>
<a NAME="E6:25_9_2_3"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <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="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">n</font>,<font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<a NAME="E7:25_9_2_3"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">n</font>,<font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E8:25_9_2_3"/><b>then </b><i><font color="Green">E200</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="heine.html#T1">HEINE:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_9_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_9_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_9_2_3_1_1"/>
( <font color="Maroon">s0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> or <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s0</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_9_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:25_9_2_3_1_1_1"/>
<font color="Maroon">s0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font>
 ;<br/><div class="add"><a NAME="E2:25_9_2_3_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/><a NAME="E3:25_9_2_3_1_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="treal_1.html#E13:25"><i><font color="Green">E65</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><a NAME="E4:25_9_2_3_1_1_1"/><b>then </b>
<font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s0</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><b>hence </b><a NAME="E5:25_9_2_3_1_1_1"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_9_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:25_9_2_3_1_1_2"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s0</font>
 ;<br/><div class="add"><a NAME="E2:25_9_2_3_1_1_2"/>
<font color="Maroon">s0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s0</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t0</font> <a href="real_1.html#K5">-</a> <font color="Maroon">s0</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K2">"</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/><a NAME="E3:25_9_2_3_1_1_2"/><b>then </b>
<font color="Maroon">s0</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E4:25_9_2_3_1_1_2"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E10:25_9_2_3"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w0</font> where <font color="Olive">w0</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w0</font> &amp; <font color="Olive">w0</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="treal_1.html#E12:25"><i><font color="Green">E64</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:25_9_2_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E56:25_9_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_9_2_4"><b>proof </b></a><div class="add">

<a NAME="E1:25_9_2_4"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="treal_1.html#D3" target="_self">Def3</a>, , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E2:25_9_2_4"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:25_9_2_4"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<font color="Maroon">d</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E57:25_9_2"/><b>then </b>
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E58:25_9_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:25_9"/>
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="txt" href="treal_1.html#E1:25"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E19:25"/>
contradiction
 <b>by </b><i>, <a class="ref" href="connsp_1.html#T11">CONNSP_1:11</a></i>;<br/>


</div>
