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

<b>let </b><font color="Maroon">V</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="metric_1.html#L1">MetrStruct</a> ;<br/>



<a NAME="E1:25"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a>  implies <font color="Maroon">V</font> is    <a href="metric_2.html#M1" target="_self">equivalence_class</a> of <font color="Maroon">M</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a> 
 ;<br/>

<a NAME="E2:25_1"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:25_1"/>
<font color="Maroon">V</font> is    <a href="metric_2.html#M1" target="_self">equivalence_class</a> of <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:25"/>
( <font color="Maroon">V</font> is    <a href="metric_2.html#M1" target="_self">equivalence_class</a> of <font color="Maroon">M</font> implies <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_2"/>
<font color="Maroon">V</font> is    <a href="metric_2.html#M1" target="_self">equivalence_class</a> of <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:25_2"/><b>then </b>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:25_2"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:25"/>
( <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a>  iff <font color="Maroon">V</font> is    <a href="metric_2.html#M1" target="_self">equivalence_class</a> of <font color="Maroon">M</font> )
 <b>by </b><i/>;<br/>


</div>
