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<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c2">z</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>
 ;<br/><b>consider </b><font color="Maroon" title="c3">a</font> being  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> , <font color="Maroon" title="c4">R</font> being   <a href="alg_1.html#NM2" title="ALG_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E3:45_1"/><i><font color="Green" title="E39">A2</font></i>: 
( <font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">a</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  &amp;  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> )
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:45_1"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="taxonom1.html#D8" target="_self" title="TAXONOM1:def.8">Def8</a></i>;<br/><a NAME="E4:45_1"/>
<font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="c3">a</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a> 
 <b>by </b><i><a class="txt" href="taxonom1.html#E3:45_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="taxonom1.html#T33" target="_self" title="TAXONOM1:th.33">Th33</a></i>;<br/><b>hence </b><a NAME="E5:45_1"/>
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="taxonom1.html#K2" title="TAXONOM1:func.2">fam_class</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E3:45_1"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="taxonom1.html#D5" target="_self" title="TAXONOM1:def.5">Def5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:45"/><b>then </b><i><font color="Green" title="E38">A3</font></i>: 
 <a href="taxonom1.html#K4" title="TAXONOM1:func.4">fam_class_metr</a> <font color="Maroon" title="c1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="taxonom1.html#K2" title="TAXONOM1:func.2">fam_class</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:45_2"/><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon" title="c2">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="taxonom1.html#K2" title="TAXONOM1:func.2">fam_class</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font>
 ;<br/><b>consider </b><font color="Maroon" title="c3">a</font> being  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> , <font color="Maroon" title="c4">R</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</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> such that </b><br/><a NAME="E3:45_2"/><i><font color="Green" title="E40">A5</font></i>: 
( <font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="c3">a</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  &amp;  <a href="eqrel_1.html#K9" title="EQREL_1:func.9">Class</a> <font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> )
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:45_2"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="taxonom1.html#D5" target="_self" title="TAXONOM1:def.5">Def5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">R1</font> = <font color="Maroon" title="c4">R</font> as   <a href="alg_1.html#NM2" title="ALG_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">M</font> ;<br/><a NAME="E5:45_2"/>
<font color="Maroon" title="c5">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c3">a</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a> 
 <b>by </b><i><a class="txt" href="taxonom1.html#E3:45_2"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="taxonom1.html#T33" target="_self" title="TAXONOM1:th.33">Th33</a></i>;<br/><b>hence </b><a NAME="E6:45_2"/>
<font color="Maroon" title="c2">z</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="txt" href="taxonom1.html#E3:45_2"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="taxonom1.html#D8" target="_self" title="TAXONOM1:def.8">Def8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:45"/><b>then </b>
 <a href="taxonom1.html#K2" title="TAXONOM1:func.2">fam_class</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:45"/>
 <a href="taxonom1.html#K4" title="TAXONOM1:func.4">fam_class_metr</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="taxonom1.html#K2" title="TAXONOM1:func.2">fam_class</a> the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E2:45"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


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