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<div class="add">

<b>let </b><font color="Maroon">M</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">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">M</font>;<br/>



<a NAME="E1:21"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">V</font> is    <a href="metric_2.html#M1" target="_self">equivalence_class</a> of <font color="Maroon">M</font> implies  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> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</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">V</font> is    <a href="metric_2.html#M1" target="_self">equivalence_class</a> of <font color="Maroon">M</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E3:21_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 <b>by </b><i/>;<br/>
<a NAME="E4:21_1"/>
<font color="Maroon">x</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">x</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:21_1"/>
 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> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:21"/>
(  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> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span> 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="21_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font><b> such that </b><a NAME="E1:21_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</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">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:21_2"/>
<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>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:21"/>
( <font color="Maroon">V</font> is    <a href="metric_2.html#M1" target="_self">equivalence_class</a> of <font color="Maroon">M</font> iff  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> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i/>;<br/>


</div>
