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

<b>let </b><font color="Maroon">M</font> be   <a href="sub_metr.html#NM1">PseudoMetricSpace</a>;<br/><b>let </b><font color="Maroon">V</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a> , <font color="Maroon">Q</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a> ;<br/><b>let </b><font color="Maroon">v</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>





<a NAME="E1:30"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">V</font>,<font color="Maroon">Q</font> <a href="metric_2.html#R4" target="_self">is_dst</a> <font color="Maroon">v</font> implies  ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">p</font>,<font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">V</font>,<font color="Maroon">Q</font> <a href="metric_2.html#R4" target="_self">is_dst</a> <font color="Maroon">v</font>
 ;<br/>

<b>consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E3:30_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 <b>by </b><i/>;<br/>
<a NAME="E4:30_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E6:30_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 <b>by </b><i/>;<br/>
<a NAME="E7:30_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:30_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 
<b>by </b><i>, <a class="ref" href="metric_2.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>hence </b><a NAME="E9:30_1"/>
 ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">p</font>,<font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="metric_2.html#D1" target="_self">Def1</a>, <a class="ref" href="metric_2.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:30"/>
(  ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">p</font>,<font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) implies <font color="Maroon">V</font>,<font color="Maroon">Q</font> <a href="metric_2.html#R4" target="_self">is_dst</a> <font color="Maroon">v</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_2"/>
 ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">p</font>,<font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 ;<br/>

<a NAME="E2:30_2"/><b>then </b>
for <font color="Olive">p1</font>, <font color="Olive">q1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Olive">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">p1</font>,<font color="Olive">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:30_2"/>
<font color="Maroon">V</font>,<font color="Maroon">Q</font> <a href="metric_2.html#R4" target="_self">is_dst</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:30"/>
( <font color="Maroon">V</font>,<font color="Maroon">Q</font> <a href="metric_2.html#R4" target="_self">is_dst</a> <font color="Maroon">v</font> iff  ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">p</font>,<font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> ) )
 <b>by </b><i/>;<br/>


</div>
