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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ,   <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> ,   <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> ] <b>means </b>$1,$2 <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> $3;<br/>
<a NAME="E1:50_1_1"/><i><font color="Green" title="E24">A1</font></i>: 
for <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 <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b3">v</font>]
 
<b>by </b><i><a class="ref" href="metric_2.html#T52" target="_self" title="METRIC_2:th.52">Th52</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E3:50_1_1"/><i><font color="Green" title="E25">A2</font></i>: 
for <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">Q</font>,<font color="Maroon" title="c2">F</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">Q</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="metric_2.html#E1:50_1_1"><i><font color="Green" title="E24">A1</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c2">F</font>
;<br/>

<b>let </b><font color="Maroon" title="c3">V</font>, <font color="Maroon" title="c4">Q</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>;<br/>



<b>assume </b><a NAME="E4:50_1_1"/><i><font color="Green" title="E26">A3</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> )
 ;<br/>

<a NAME="E5:50_1_1"/>
<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">Q</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Maroon" title="c2">F</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">Q</font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E3:50_1_1"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:50_1_1"/>
<font color="Maroon" title="c2">F</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font>
 <b>by </b><i><a class="txt" href="metric_2.html#E4:50_1_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="metric_2.html#D5" target="_self" title="METRIC_2:def.5">Def5</a></i>;<br/>


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