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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="sub_metr.html#NM1">PseudoMetricSpace</a>;<br/>

<a NAME="E1:47"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="metric_2.html#K2" target="_self">-neighbour</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="metric_2.html#K2" target="_self">-neighbour</a> </span>)</span>,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K3">:]</a></span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_2.html#K9" target="_self">set_in_rel</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="metric_2.html#K6" target="_self">elem_in_rel_1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K7" target="_self">elem_in_rel_2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K5" target="_self">real_in_rel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="mcart_1.html#K13">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="metric_2.html#K2" target="_self">-neighbour</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="metric_2.html#K2" target="_self">-neighbour</a> </span>)</span>,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K3">:]</a></span>;<br/>

<b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_2.html#K9" target="_self">set_in_rel</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_2.html#K2" target="_self">-neighbour</a> , <font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:47_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="mcart_1.html#K3">]</a></span> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="metric_2.html#R4" target="_self">is_dst</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="metric_2.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E4:47_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_2.html#K6" target="_self">elem_in_rel_1</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E3:47_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E5:47_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_2.html#K7" target="_self">elem_in_rel_2</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E3:47_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E6:47_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_2.html#K5" target="_self">real_in_rel</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E3:47_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:47_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="metric_2.html#K6" target="_self">elem_in_rel_1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K7" target="_self">elem_in_rel_2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K5" target="_self">real_in_rel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="mcart_1.html#K13">:]</a></span>
 <b>by </b><i><a class="txt" href="metric_2.html#E3:47_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="metric_2.html#E4:47_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="metric_2.html#E5:47_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="mcart_1.html#T73">MCART_1:73</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:47"/>
 <a href="metric_2.html#K9" target="_self">set_in_rel</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="mcart_1.html#K13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="metric_2.html#K6" target="_self">elem_in_rel_1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K7" target="_self">elem_in_rel_2</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K5" target="_self">real_in_rel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="mcart_1.html#K13">:]</a></span>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>


</div>
