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<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="tbsp_1.html#V5" title="TBSP_1:attr.5">bounded</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="taxonom1.html#K4" title="TAXONOM1:func.4">fam_class_metr</a> <font color="Maroon" title="c1">M</font> is    <a href="taxonom1.html#M2" title="TAXONOM1:mode.2">Strong_Classification</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span><br/>

<a NAME="E1:54"/><i><font color="Green" title="E44">A1</font></i>: 
 <a href="taxonom1.html#K4" title="TAXONOM1:func.4">fam_class_metr</a> <font color="Maroon" title="c1">M</font> is    <a href="taxonom1.html#M1" title="TAXONOM1:mode.1">Classification</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="taxonom1.html#T42" target="_self" title="TAXONOM1:th.42">Th42</a></i>;<br/>
<a NAME="E2:54"/><i><font color="Green" title="E45">A2</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="taxonom1.html#K4" title="TAXONOM1:func.4">fam_class_metr</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="taxonom1.html#T41" target="_self" title="TAXONOM1:th.41">Th41</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c2">a</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as  non <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<a NAME="E4:54"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a>
 
<b>by </b><i><a class="ref" href="metric_1.html#D7" title="METRIC_1:def.7">METRIC_1:def 7</a></i>;<br/>
<a NAME="E5:54"/><b>then </b>
 <a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">a</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="taxonom1.html#T16" target="_self" title="TAXONOM1:th.16">Th16</a></i>;<br/>
<a NAME="E6:54"/><b>then </b><i><font color="Green" title="E46">A3</font></i>: 
 <a href="taxonom1.html#K3" title="TAXONOM1:func.3">dist_toler</a> <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">a</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="taxonom1.html#T33" target="_self" title="TAXONOM1:th.33">Th33</a></i>;<br/>
<a NAME="E7:54"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a>
 
<b>by </b><i><a class="ref" href="metric_1.html#D9" title="METRIC_1:def.9">METRIC_1:def 9</a></i>;<br/>
<a NAME="E8:54"/><b>then </b>
 <a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">a</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="taxonom1.html#T17" target="_self" title="TAXONOM1:th.17">Th17</a></i>;<br/>
<a NAME="E9:54"/><b>then </b>
 <a href="taxonom1.html#K3" title="TAXONOM1:func.3">dist_toler</a> <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">a</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="taxonom1.html#T33" target="_self" title="TAXONOM1:th.33">Th33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">R</font> = <span class="p1">(<span class="default"><a href="taxonom1.html#K3" title="TAXONOM1:func.3">dist_toler</a> <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">a</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  as   <a href="alg_1.html#NM2" title="ALG_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="txt" href="#E6:54"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="taxonom1.html#T9" target="_self" title="TAXONOM1:th.9">Th9</a></i>;<br/>
<a NAME="E11:54"/>
 <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="taxonom1.html#K4" title="TAXONOM1:func.4">fam_class_metr</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="taxonom1.html#D8" target="_self" title="TAXONOM1:def.8">Def8</a></i>;<br/>
<a NAME="E12:54"/><b>then </b>
 <a href="eqrel_1.html#K10" title="EQREL_1:func.10">SmallestPartition</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="taxonom1.html#K4" title="TAXONOM1:func.4">fam_class_metr</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="taxonom1.html#T36" target="_self" title="TAXONOM1:th.36">Th36</a></i>;<br/>
<b>hence </b><a NAME="E13:54"/>
 <a href="taxonom1.html#K4" title="TAXONOM1:func.4">fam_class_metr</a> <font color="Maroon" title="c1">M</font> is    <a href="taxonom1.html#M2" title="TAXONOM1:mode.2">Strong_Classification</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="txt" href="#E1:54"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="#E2:54"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="taxonom1.html#D2" target="_self" title="TAXONOM1:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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