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<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  <a href="numbers.html#K1">REAL</a> ;<br/>



<a NAME="E1:37"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="metric_2.html#K5" target="_self">real_in_rel</a> <font color="Maroon">M</font> implies  ex <font color="Olive">V</font>, <font color="Olive">Q</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a>  st <font color="Olive">V</font>,<font color="Olive">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="37_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="metric_2.html#K5" target="_self">real_in_rel</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:37_1"/><b>then </b>
<font color="Maroon">v</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="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  :  ex <font color="Olive">V</font>, <font color="Olive">Q</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a>  st <font color="Olive">V</font>,<font color="Olive">Q</font> <a href="metric_2.html#R4" target="_self">is_dst</a> <font color="Olive">r</font> </span>}</span> 
 
;<br/>
<a NAME="E3:37_1"/><b>then </b>
 ex <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> &amp;  ex <font color="Olive">V</font>, <font color="Olive">Q</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a>  st <font color="Olive">V</font>,<font color="Olive">Q</font> <a href="metric_2.html#R4" target="_self">is_dst</a> <font color="Olive">r</font> )
 
;<br/>
<b>hence </b><a NAME="E4:37_1"/>
 ex <font color="Olive">V</font>, <font color="Olive">Q</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a>  st <font color="Olive">V</font>,<font color="Olive">Q</font> <a href="metric_2.html#R4" target="_self">is_dst</a> <font color="Maroon">v</font>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:37"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="metric_2.html#K5" target="_self">real_in_rel</a> <font color="Maroon">M</font> iff  ex <font color="Olive">V</font>, <font color="Olive">Q</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a>  st <font color="Olive">V</font>,<font color="Olive">Q</font> <a href="metric_2.html#R4" target="_self">is_dst</a> <font color="Maroon">v</font> )
 <b>by </b><i/>;<br/>


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