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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:21"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is    <a href="metric_2.html#M1" target="_self">equivalence_class</a> of <font color="Maroon">c<sub>1</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1"/>
<font color="Maroon">c<sub>2</sub></font> is    <a href="metric_2.html#M1" target="_self">equivalence_class</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:21_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 <b>by </b><i><a class="ref" href="metric_2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E4:21_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="metric_2.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E5:21_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="metric_2.html#E3:21_1"><i><font color="Green">E15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:21"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> implies <font color="Maroon">c<sub>2</sub></font> is    <a href="metric_2.html#M1" target="_self">equivalence_class</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:21_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:21_2"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 
<b>by </b><i><a class="ref" href="metric_2.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E3:21_2"/>
<font color="Maroon">c<sub>2</sub></font> is    <a href="metric_2.html#M1" target="_self">equivalence_class</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="metric_2.html#E1:21_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="metric_2.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:21"/>
( <font color="Maroon">c<sub>2</sub></font> is    <a href="metric_2.html#M1" target="_self">equivalence_class</a> of <font color="Maroon">c<sub>1</sub></font> iff ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="metric_2.html#E1:21"><i><font color="Green">E14</font></i></a></i>;<br/>


</div>
