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<b>let </b><font color="Maroon" title="c1">M</font> be   <a href="sub_metr.html#NM1" title="SUB_METR:NM.1">PseudoMetricSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">VQv</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a> </span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>;<br/>



<a NAME="E1:45"/>
( <font color="Maroon" title="c2">VQv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_2.html#K9" title="METRIC_2:func.9">set_in_rel</a> <font color="Maroon" title="c1">M</font> implies  ex <font color="Olive" title="b1">V</font>, <font color="Olive" title="b2">Q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a>  ex <font color="Olive" title="b3">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c2">VQv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b3">v</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">Q</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Olive" title="b3">v</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1"/>
<font color="Maroon" title="c2">VQv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_2.html#K9" title="METRIC_2:func.9">set_in_rel</a> <font color="Maroon" title="c1">M</font>
 ;<br/>

<a NAME="E2:45_1"/><b>then </b>
 ex <font color="Olive" title="b1">S</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a> </span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> st <br/>( <font color="Maroon" title="c2">VQv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">S</font> &amp;  ex <font color="Olive" title="b2">V</font>, <font color="Olive" title="b3">Q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a>  ex <font color="Olive" title="b4">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b2">V</font>,<font color="Olive" title="b3">Q</font>,<font color="Olive" title="b4">v</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; <font color="Olive" title="b2">V</font>,<font color="Olive" title="b3">Q</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Olive" title="b4">v</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:45_1"/>
 ex <font color="Olive" title="b1">V</font>, <font color="Olive" title="b2">Q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a>  ex <font color="Olive" title="b3">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c2">VQv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b3">v</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">Q</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Olive" title="b3">v</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:45"/>
( <font color="Maroon" title="c2">VQv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_2.html#K9" title="METRIC_2:func.9">set_in_rel</a> <font color="Maroon" title="c1">M</font> iff  ex <font color="Olive" title="b1">V</font>, <font color="Olive" title="b2">Q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a>  ex <font color="Olive" title="b3">v</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c2">VQv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b3">v</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">Q</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Olive" title="b3">v</font> ) )
 ;<br/>


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