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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>









<b>assume </b><a NAME="E1:20"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<a NAME="E2:20"/><i><font color="Green">E19</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:20"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>8</sub></font> st <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>;<br/>

<b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E1:20_1"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5a.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan5a.html#E19"><i><font color="Green">Lemma16</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_1"/>
(  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_1_1"/><i><font color="Green">E21</font></i>: 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:20_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5a.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan5a.html#E1:20_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E3:20_1_1_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5a.html#E1:20_1_1_1_1"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E4:20_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E5:20_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5a.html#E19"><i><font color="Green">Lemma16</font></i></a></i>;<br/><b>hence </b><a NAME="E6:20_1_1_1_1"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:20_1_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan5a.html#E3:20_1_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_1_2"/><i><font color="Green">E21</font></i>: 
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:20_1_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5a.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan5a.html#E1:20_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E3:20_1_1_1_2"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a></i>;<br/><a NAME="E4:20_1_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E5:20_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5a.html#E19"><i><font color="Green">Lemma16</font></i></a></i>;<br/><b>hence </b><a NAME="E6:20_1_1_1_2"/>
 <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:20_1_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan5a.html#E3:20_1_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> as    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="ref" href="rcomp_1.html#D7">RCOMP_1:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E4:20_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#T38">RCOMP_1:38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> ;<br/>
<a NAME="E6:20_1"/><i><font color="Green">E21</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/>
<a NAME="E7:20_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_2"/><i><font color="Green">E22</font></i>: 
(  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="square_1.html#K1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>14</sub></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="E2:20_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E3:20_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:20_1_2"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> )
 ;<br/>
<a NAME="E6:20_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:20_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E7:20_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E8:20_1_2"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E5:20_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:20_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:20_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E10:20_1_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E5:20_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:20_1_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan5a.html#E8:20_1_2"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:20_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="txt" href="jordan5a.html#E5:20_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:20_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E6:20_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:20_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> 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> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>;<br/>
<a NAME="E10:20_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E4:20_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan5a.html#E8:20_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<i><font color="Green">E25</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> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="topmetr.html#D8">TOPMETR:def 8</a></i>
;<br/>
<i><font color="Green">E26</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> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="topmetr.html#D8">TOPMETR:def 8</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan5a.html#E4:20_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan5a.html#E9:20_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan5a.html#E11:20_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E15:20_1"/>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <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> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#D8">TOPMETR:def 8</a></i>;<br/>
<a NAME="E16:20_1"/><b>then </b><i><font color="Green">E27</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E17:20_1"/>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <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> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#D8">TOPMETR:def 8</a></i>;<br/>
<a NAME="E18:20_1"/><b>then </b><i><font color="Green">E28</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E19:20_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="ref" href="jordan5a.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E20:20_1"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5a.html#E16:20_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a>, <a class="ref" href="rcomp_1.html#T37">RCOMP_1:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>13</sub></font> <b>by </b><i><a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E23:20_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tmap_1.html#D2">TMAP_1:def 2</a></i>;<br/>
<a NAME="E24:20_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E4:20_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan5a.html#E23:20_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E26:20_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="ref" href="connsp_2.html#T8">CONNSP_2:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="txt" href="jordan5a.html#E12:20_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E28:20_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> 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> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E12:20_1"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E29:20_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>;<br/>
<a NAME="E30:20_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K3">Family_open_set</a> <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E18:20_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan5a.html#E26:20_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="jordan5a.html#E29:20_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E32:20_1"/>
( not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1_5"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_5_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan5a.html#E1:20_1_5"><i><font color="Green">E34</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_5_1_1"><b>suppose </b></a><a NAME="E1:20_1_5_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><div class="add"><a NAME="E2:20_1_5_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan5a.html#E26:20_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>hence </b><a NAME="E3:20_1_5_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5a.html#E10:20_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan5a.html#E23:20_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan5a.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_5_1_2"><b>suppose </b></a><a NAME="E1:20_1_5_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><div class="add"><a NAME="E2:20_1_5_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan5a.html#E26:20_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>hence </b><a NAME="E3:20_1_5_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5a.html#E10:20_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan5a.html#E23:20_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan5a.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E33:20_1"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> is <a href="rcomp_1.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5a.html#E30:20_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan5a.html#T11" target="_self">Th11</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E35:20_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5a.html#E26:20_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="rcomp_1.html#T39">RCOMP_1:39</a></i>;<br/>
<a NAME="E36:20_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E26:20_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan5a.html#E35:20_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E37:20_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E38:20_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5a.html#E24:20_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E39:20_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>8</sub></font> st <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:20"/>
<font color="Maroon">c<sub>7</sub></font> <a href="fcont_1.html#R1">is_continuous_in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5a.html#E2:20"><i><font color="Green">E19</font></i></a>, <a class="ref" href="fcont_1.html#T5">FCONT_1:5</a></i>;<br/>


</div>
