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<b>let </b><font color="Maroon" title="c1">V</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ;<br/>



<a NAME="E1:25"/><i><font color="Green" title="E16">A1</font></i>: 
( <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a>  implies <font color="Maroon" title="c1">V</font> is    <a href="metric_2.html#M1" title="METRIC_2:mode.1">equivalence_class</a> of <font color="Maroon" title="c2">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" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a> 
 ;<br/>

<a NAME="E2:25_1"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font> st <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</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/>
<b>hence </b><a NAME="E3:25_1"/>
<font color="Maroon" title="c1">V</font> is    <a href="metric_2.html#M1" title="METRIC_2:mode.1">equivalence_class</a> of <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="metric_2.html#D3" target="_self" title="METRIC_2:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:25"/>
( <font color="Maroon" title="c1">V</font> is    <a href="metric_2.html#M1" title="METRIC_2:mode.1">equivalence_class</a> of <font color="Maroon" title="c2">M</font> implies <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-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" title="c1">V</font> is    <a href="metric_2.html#M1" title="METRIC_2:mode.1">equivalence_class</a> of <font color="Maroon" title="c2">M</font>
 ;<br/>

<a NAME="E2:25_2"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font> st <font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</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#D3" target="_self" title="METRIC_2:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E3:25_2"/>
<font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:25"/>
( <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a>  iff <font color="Maroon" title="c1">V</font> is    <a href="metric_2.html#M1" title="METRIC_2:mode.1">equivalence_class</a> of <font color="Maroon" title="c2">M</font> )
 <b>by </b><i><a class="txt" href="metric_2.html#E1:25"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/>


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