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<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="alg_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:46"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K3" target="_self">dist_toler</a> <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="lang1.html#K17">[*]</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:46_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/><b>hence </b><a NAME="E2:46_1"/>
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:46"/><b>then </b><i><font color="Green">E40</font></i>: 
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="taxonom1.html#V1" target="_self">nonnegative</a>
 
<b>by </b><i><a class="ref" href="taxonom1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:46"/><i><font color="Green">E41</font></i>: 
( the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="metric_1.html#V2">Reflexive</a> &amp; the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="metric_1.html#V3">discerning</a> )
 
<b>by </b><i><a class="ref" href="metric_1.html#D7">METRIC_1:def 7</a>, <a class="ref" href="metric_1.html#D8">METRIC_1:def 8</a></i>;<br/>
<a NAME="E5:46"/>
<span class="p1">(<span class="default"><a href="taxonom1.html#K1" target="_self">low_toler</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">c<sub>1</sub></font>,0</span>)</span> <a href="lang1.html#K17">[*]</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="taxonom1.html#E1:46"><i><font color="Green">E39</font></i></a>, <a class="ref" href="taxonom1.html#T33" target="_self">Th33</a></i>;<br/>
<b>hence </b><a NAME="E6:46"/>
 <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K11">SmallestPartition</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E3:46"><i><font color="Green">E40</font></i></a>, <a class="txt" href="taxonom1.html#E4:46"><i><font color="Green">E41</font></i></a>, <a class="ref" href="taxonom1.html#T25" target="_self">Th25</a></i>;<br/>


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