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<b>set </b><font color="Maroon" title="c2">Z</font> =  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <span class="p1">(<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="p1">(<span class="default"><a href="metric_2.html#K10" title="METRIC_2:func.10">nbourdist</a> <font color="Maroon" title="c1">M</font></span>)</span> #);<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">V</font>, <font color="Maroon" title="c4">Q</font>, <font color="Maroon" title="c5">W</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <span class="p1">(<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="p1">(<span class="default"><a href="metric_2.html#K10" title="METRIC_2:func.10">nbourdist</a> <font color="Maroon" title="c1">M</font></span>)</span> #);<br/><b>reconsider </b><font color="Maroon" title="c6">V'</font> = <font color="Maroon" title="c3">V</font>, <font color="Maroon" title="c7">Q'</font> = <font color="Maroon" title="c4">Q</font>, <font color="Maroon" title="c8">W'</font> = <font color="Maroon" title="c5">W</font> as    <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/><a NAME="E2:54_1_1_1"/><i><font color="Green" title="E28">A1</font></i>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</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> <span class="p1">(<span class="default"><a href="metric_2.html#K10" title="METRIC_2:func.10">nbourdist</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">V'</font>,<font color="Maroon" title="c7">Q'</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><a NAME="E3:54_1_1_1"/><i><font color="Green" title="E29">A2</font></i>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c4">Q</font>,<font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_2.html#K10" title="METRIC_2:func.10">nbourdist</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">Q'</font>,<font color="Maroon" title="c6">V'</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><a NAME="E4:54_1_1_1"/><i><font color="Green" title="E30">A3</font></i>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_2.html#K10" title="METRIC_2:func.10">nbourdist</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">V'</font>,<font color="Maroon" title="c8">W'</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><a NAME="E5:54_1_1_1"/><i><font color="Green" title="E31">A4</font></i>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c4">Q</font>,<font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_2.html#K10" title="METRIC_2:func.10">nbourdist</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">Q'</font>,<font color="Maroon" title="c8">W'</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><b>thus </b><a NAME="E6:54_1_1_1"/>
(  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</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="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q</font> )
 <b>by </b><i><a class="txt" href="metric_2.html#E2:54_1_1_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="metric_2.html#T54" target="_self" title="METRIC_2:th.54">Th54</a></i>;<br/><b>thus </b><a NAME="E7:54_1_1_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c4">Q</font>,<font color="Maroon" title="c3">V</font>
 <b>by </b><i><a class="txt" href="metric_2.html#E2:54_1_1_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="metric_2.html#E3:54_1_1_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="metric_2.html#T55" target="_self" title="METRIC_2:th.55">Th55</a></i>;<br/><b>thus </b><a NAME="E8:54_1_1_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c5">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">Q</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c4">Q</font>,<font color="Maroon" title="c5">W</font></span>)</span>
 <b>by </b><i><a class="txt" href="metric_2.html#E2:54_1_1_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="metric_2.html#E4:54_1_1_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="metric_2.html#E5:54_1_1_1"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="metric_2.html#T56" target="_self" title="METRIC_2:th.56">Th56</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:54_1_1"/>
 <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <span class="p1">(<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="p1">(<span class="default"><a href="metric_2.html#K10" title="METRIC_2:func.10">nbourdist</a> <font color="Maroon" title="c1">M</font></span>)</span> #) is   <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>
 <b>by </b><i><a class="ref" href="metric_1.html#T6" title="METRIC_1:th.6">METRIC_1:6</a></i>;<br/>


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