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<b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a> <a href="tbsp_1.html#V5">bounded</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/>

<a NAME="E1:54"/><i><font color="Green">E29</font></i>: 
 <a href="taxonom1.html#K4" target="_self">fam_class_metr</a> <font color="Maroon">M</font> is    <a href="taxonom1.html#M1" target="_self">Classification</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:54"/><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K4" target="_self">fam_class_metr</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = 0 as  non <a href="xxreal_0.html#V3">negative</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<a NAME="E4:54"/>
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> is <a href="metric_1.html#V2">Reflexive</a>
 
<b>by </b><i><a class="ref" href="metric_1.html#D7">METRIC_1:def 7</a></i>;<br/>
<a NAME="E5:54"/><b>then </b>
 <a href="taxonom1.html#K1" target="_self">low_toler</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font>,<font color="Maroon">a</font> <a href="relat_2.html#R1">is_reflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:54"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="taxonom1.html#K3" target="_self">dist_toler</a> <font color="Maroon">M</font>,<font color="Maroon">a</font> <a href="relat_2.html#R1">is_reflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:54"/>
the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> is <a href="metric_1.html#V4">symmetric</a>
 
<b>by </b><i><a class="ref" href="metric_1.html#D9">METRIC_1:def 9</a></i>;<br/>
<a NAME="E8:54"/><b>then </b>
 <a href="taxonom1.html#K1" target="_self">low_toler</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font>,<font color="Maroon">a</font> <a href="relat_2.html#R3">is_symmetric_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:54"/><b>then </b>
 <a href="taxonom1.html#K3" target="_self">dist_toler</a> <font color="Maroon">M</font>,<font color="Maroon">a</font> <a href="relat_2.html#R3">is_symmetric_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">R</font> = <span class="p1">(<span class="default"><a href="taxonom1.html#K3" target="_self">dist_toler</a> <font color="Maroon">M</font>,<font color="Maroon">a</font></span>)</span> <a href="lang1.html#K17">[*]</a>  as   <a href="alg_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">M</font> <b>by </b><i>, </i>;<br/>
<a NAME="E11:54"/>
 <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K4" target="_self">fam_class_metr</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:54"/><b>then </b>
 <a href="eqrel_1.html#K11">SmallestPartition</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K4" target="_self">fam_class_metr</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E13:54"/>
 <a href="taxonom1.html#K4" target="_self">fam_class_metr</a> <font color="Maroon">M</font> is    <a href="taxonom1.html#M2" target="_self">Strong_Classification</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 <b>by </b><i>, , </i>;<br/>


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