<?xml version="1.0"?>
<div class="add">

<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>, <font color="Olive" title="b3">W</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="b3">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<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="b3">W</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">V</font>, <font color="Maroon" title="c3">Q</font>, <font color="Maroon" title="c4">W</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="c4">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<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="c4">W</font></span>)</span></span><br/>



<b>consider </b><font color="Maroon" title="c5">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="E2:53"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="E3:53"/><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c5">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#E2:53"><i><font color="Green" title="E28">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="c6">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="E5:53"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="E6:53"/><i><font color="Green" title="E31">A4</font></i>: 
<font color="Maroon" title="c6">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#E5:53"><i><font color="Green" title="E30">A3</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="c7">w</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="E8:53"/><i><font color="Green" title="E32">A5</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">w</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="E9:53"/><i><font color="Green" title="E33">A6</font></i>: 
<font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">W</font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E8:53"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="metric_2.html#T9" target="_self" title="METRIC_2:th.9">Th9</a></i>;<br/>
<a NAME="E10:53"/><i><font color="Green" title="E34">A7</font></i>: 
<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>  <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#E3:53"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="metric_2.html#E6:53"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="metric_2.html#D13" target="_self" title="METRIC_2:def.13">Def13</a></i>;<br/>
<a NAME="E11:53"/><i><font color="Green" title="E35">A8</font></i>: 
<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="c4">W</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="c7">w</font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E3:53"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="metric_2.html#E9:53"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="metric_2.html#D13" target="_self" title="METRIC_2:def.13">Def13</a></i>;<br/>
<a NAME="E12:53"/>
<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="c4">W</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="c6">q</font>,<font color="Maroon" title="c7">w</font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E6:53"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="metric_2.html#E9:53"><i><font color="Green" title="E33">A6</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="E13:53"/>
<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="c4">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<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="c4">W</font></span>)</span>
 <b>by </b><i><a class="txt" href="metric_2.html#E10:53"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="metric_2.html#E11:53"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
