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

<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   <a href="metric_1.html#NM2" title="METRIC_1:NM.2">PseudoMetricSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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 <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="Olive" title="b1">V</font>,<font color="Olive" title="b2">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="Olive" title="b1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Q</font> )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>  <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> )</span><br/>



<a NAME="E1:51"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
( <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_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E2:51_1"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_2.html#T15" target="_self" title="METRIC_2:th.15">Th23</a></span>;<br/>
<a NAME="E3:51_1"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="metric_2.html#T4" target="_self" title="METRIC_2:th.4">Th9</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:51_1"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_2.html#T15" target="_self" title="METRIC_2:th.15">Th23</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:51_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E7:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E5:51_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="metric_2.html#T2" target="_self" title="METRIC_2:th.2">Th7</a></span>;<br/>
<a NAME="E8:51_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="metric_2.html#T4" target="_self" title="METRIC_2:th.4">Th9</a></span>;<br/>
<span class="kw">then </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> <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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="metric_2.html#D13" target="_self" title="METRIC_2:def.13">Def13</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="metric_2.html#D1" target="_self" title="METRIC_2:def.1">Def1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:51_1"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:51"/>
( <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>  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_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:51_2"/><span class="lab"><font color="Green" title="E27">A6</font></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> <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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Q</font></span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:51_2"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_2.html#T15" target="_self" title="METRIC_2:th.15">Th23</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:51_2"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_2.html#T15" target="_self" title="METRIC_2:th.15">Th23</a></span>;<br/>
<a NAME="E6:51_2"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_2"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="metric_2.html#T4" target="_self" title="METRIC_2:th.4">Th9</a></span>;<br/>
<a NAME="E7:51_2"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_2"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="metric_2.html#T4" target="_self" title="METRIC_2:th.4">Th9</a></span>;<br/>
<a NAME="E8:51_2"/><span class="kw">then </span>
 <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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_2"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E6:51_2"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="metric_2.html#D13" target="_self" title="METRIC_2:def.13">Def13</a></span>;<br/>
<a NAME="E9:51_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_2.html#D1" target="_self" title="METRIC_2:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:51_2"/>
<font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_2"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E5:51_2"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="metric_2.html#T8" target="_self" title="METRIC_2:th.8">Th13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E26">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
