<?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>;<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> ;<br/>



<a NAME="E1:51"/><i><font color="Green" title="E26">A1</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="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1"/><i><font color="Green" title="E27">A2</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="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<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="E3:51_1"/><i><font color="Green" title="E28">A3</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="E4:51_1"/><i><font color="Green" title="E29">A4</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#E3:51_1"><i><font color="Green" title="E28">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="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="E6:51_1"/><i><font color="Green" title="E30">A5</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="E7:51_1"/>
<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#E6:51_1"><i><font color="Green" title="E30">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="E8:51_1"/><b>then </b>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="metric_2.html#E1:51_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="metric_2.html#E4:51_1"><i><font color="Green" title="E29">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="E9:51_1"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="metric_2.html#R3" title="METRIC_2:pred.3">tolerates</a> <font color="Maroon" title="c5">q</font>
 
<b>by </b><i><a class="ref" href="metric_2.html#D1" target="_self" title="METRIC_2:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E10:51_1"/>
<font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Q</font>
 <b>by </b><i><a class="txt" href="metric_2.html#E3:51_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="metric_2.html#E6:51_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="metric_2.html#T13" target="_self" title="METRIC_2:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:51"/>
( <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Q</font> implies <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="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_2"/><i><font color="Green" title="E27">A6</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Q</font>
 ;<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="E3:51_2"/><i><font color="Green" title="E28">A7</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="E4:51_2"/><i><font color="Green" title="E29">A8</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#E3:51_2"><i><font color="Green" title="E28">A7</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="E6:51_2"/><i><font color="Green" title="E30">A9</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="E7:51_2"/><i><font color="Green" title="E31">A10</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="metric_2.html#R3" title="METRIC_2:pred.3">tolerates</a> <font color="Maroon" title="c5">q</font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E1:51_2"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="metric_2.html#E4:51_2"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="metric_2.html#E6:51_2"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="metric_2.html#T7" target="_self" title="METRIC_2:th.7">Th7</a></i>;<br/>
<a NAME="E8:51_2"/>
<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#E6:51_2"><i><font color="Green" title="E30">A9</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="c4">p</font>,<font color="Maroon" title="c5">q</font>
<b>by </b><i><a class="txt" href="metric_2.html#E4:51_2"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="metric_2.html#D13" target="_self" title="METRIC_2:def.13">Def13</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="metric_2.html#E7:51_2"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="metric_2.html#D1" target="_self" title="METRIC_2:def.1">Def1</a></i>
;<br/>
<b>hence </b><a NAME="E10:51_2"/>
<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="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:51"/>
( <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="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Q</font> )
 <b>by </b><i><a class="txt" href="metric_2.html#E1:51"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>


</div>
