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

<b>thus </b><a NAME="E1:38_1_1"/>
( <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/> <a href="metric_1.html#K2">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> ) &amp; ( for <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  st ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/> <a href="metric_1.html#K2">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> ) holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="real_1.html#NM1">Real</a>] <b>means </b>for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/> <a href="metric_1.html#K2">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">I</font> = <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font> );<br/>
<b>set </b><font color="Maroon">J</font> = <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">r</font>] </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:38_1_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E4:38_1_1_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/> <a href="metric_1.html#K2">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:38_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">c</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:38_1_1_1"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T1">METRIC_1:1</a></i>;<br/>
<a NAME="E9:38_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">r</font>] </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E10:38_1_1_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">r</font>] </span>}</span>  &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] </span>}</span>  holds <br/><font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:38_1_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">r</font>] </span>}</span> 
 <b>and </b><br/><a NAME="E2:38_1_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">t</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:38_1_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E5:38_1_1_1_1"/><i><font color="Green">E64</font></i>: 
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:38_1_1_1_1"/>
 ex <font color="Olive">t'</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/> <a href="metric_1.html#K2">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t'</font> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:38_1_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:38_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">I</font> = <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E13:38_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">r</font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">J</font> = <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">r</font>] </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>consider </b><font color="Maroon">d</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="E16:38_1_1_1"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">a</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> holds <br/><font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E17:38_1_1_1"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">b</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b</font>
 <b>by </b><i>, , , <a class="ref" href="real_2.html#T204">REAL_2:204</a></i>;<br/>
<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>thus </b><a NAME="E18:38_1_1_1"/>
<font color="Maroon">d</font> is   <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>thus </b><a NAME="E19:38_1_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/> <a href="metric_1.html#K2">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font>;<br/>

<b>assume </b><a NAME="E1:38_1_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E2:38_1_1_1_2"/><b>then </b>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 
;<br/>
<b>hence </b><a NAME="E3:38_1_1_1_2"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">s</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E20:38_1_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/> <a href="metric_1.html#K2">dist</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E21:38_1_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
;<br/>
<b>hence </b><a NAME="E22:38_1_1_1"/>
<font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:38_1_1"/>
( not <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>


</div>
