<?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> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <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>;<br/>





<b>assume </b><a NAME="E1:18"/><i><font color="Green">E16</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>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E2:18"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="rcomp_1.html#V3">open</a>
 ;<br/>

<a NAME="E3:18"/>
for <font color="Olive">b<sub>1</sub></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> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</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>;<br/>

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jordan5a.html#E1:18"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="rcomp_1.html#M1">Neighbourhood</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E4:18_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan5a.html#E2:18"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan5a.html#E1:18_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="rcomp_1.html#T42">RCOMP_1:42</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></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="E6:18_1"/><i><font color="Green">E20</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> )
 <b>by </b><i><a class="ref" href="rcomp_1.html#D7">RCOMP_1:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E8:18_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:18_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</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="E1:18_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<a NAME="E2:18_1_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E4:18_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jordan5a.html#E1:18_1_1_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</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#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan5a.html#E2:18_1_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<i><font color="Green">E23</font></i>:  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font> = 
the <a href="metric_1.html#U1">distance</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> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
<a href="metric_1.html#K7">real_dist</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#D1">TOPMETR:def 1</a></i>
;<br/>
<a NAME="E6:18_1_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E6:18_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan5a.html#E1:18_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>
<a NAME="E7:18_1_1_1"/><b>then </b>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a></i>;<br/>
<a NAME="E8:18_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E5:18_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="metric_1.html#T10">METRIC_1:10</a></i>;<br/>
<b>hence </b><a NAME="E9:18_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>


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

<b>thus </b><a NAME="E2:18_1_1"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</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="E1:18_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E2:18_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</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="struct_0.html#NM1">Element</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> :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></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> <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="E4:18_1_1_2"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/>
<a NAME="E5:18_1_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>11</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#K2">Closed-Interval-MSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
;<br/>
<a NAME="E6:18_1_1_2"/><i><font color="Green">E23</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> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="metric_1.html#K8">RealSpace</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#D1">TOPMETR:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jordan5a.html#E4:18_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan5a.html#E5:18_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="jordan5a.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan5a.html#E1:18_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5a.html#E5:18_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan5a.html#E6:18_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font> = 
the <a href="metric_1.html#U1">distance</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> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
<a href="metric_1.html#K7">real_dist</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#D1">TOPMETR:def 1</a></i>
;<br/>
<a NAME="E10:18_1_1_2"/><b>then </b>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E4:18_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="metric_1.html#T10">METRIC_1:10</a></i>;<br/>
<a NAME="E11:18_1_1_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5a.html#E4:18_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E12:18_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="jordan5a.html#E6:18_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="rcomp_1.html#T8">RCOMP_1:8</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:18_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="jordan5a.html#E1:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan5a.html#E4:18_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan5a.html#E6:18_1"><i><font color="Green">E20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:18"/>
<font color="Maroon">c<sub>4</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="ref" href="pcomps_1.html#D5">PCOMPS_1:def 5</a></i>;<br/>


</div>
