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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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  <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="Olive" title="b1">V</font>,<font color="Olive" title="b2">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="Olive" title="b2">Q</font>,<font color="Olive" title="b1">V</font></span><br/><b>let </b><font color="Maroon" title="c2">V</font>, <font color="Maroon" title="c3">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="c2">V</font>,<font color="Maroon" title="c3">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="c3">Q</font>,<font color="Maroon" title="c2">V</font></span><br/>



<b>consider </b><font color="Maroon" title="c4">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E2:52"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> 
 <b>by </b><i><a class="ref" href="metric_2.html#T23" target="_self" title="METRIC_2:th.23">Th23</a></i>;<br/>
<a NAME="E3:52"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E2:52"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="metric_2.html#T9" target="_self" title="METRIC_2:th.9">Th9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E5:52"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> 
 <b>by </b><i><a class="ref" href="metric_2.html#T23" target="_self" title="METRIC_2:th.23">Th23</a></i>;<br/>
<a NAME="E6:52"/><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E5:52"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="metric_2.html#T9" target="_self" title="METRIC_2:th.9">Th9</a></i>;<br/>
<b>then </b><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="c2">V</font>,<font color="Maroon" title="c3">Q</font> = 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c4">p</font>
<b>by </b><i><a class="txt" href="metric_2.html#E3:52"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="metric_2.html#D13" target="_self" title="METRIC_2:def.13">Def13</a></i>
<br/>.= 
<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="c3">Q</font>,<font color="Maroon" title="c2">V</font>
<b>by </b><i><a class="txt" href="metric_2.html#E3:52"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="metric_2.html#E6:52"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="metric_2.html#D13" target="_self" title="METRIC_2:def.13">Def13</a></i>
;<br/>
<b>hence </b><a NAME="E8:52"/>
<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="c2">V</font>,<font color="Maroon" title="c3">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="c3">Q</font>,<font color="Maroon" title="c2">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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