<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#V6">Reflexive</a> <a href="metric_1.html#V8">symmetric</a>  <a href="metric_1.html#L1">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">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K4" target="_self">fam_class_metr</a> <font color="Maroon">M</font>
 ;<br/><b>consider </b><font color="Maroon">a</font> being  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> , <font color="Maroon">R</font> being   <a href="alg_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E3:45_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">R</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">M</font>,<font color="Maroon">a</font></span>)</span> <a href="lang1.html#K17">[*]</a>  &amp;  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:45_1"/>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <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">M</font>,<font color="Maroon">a</font></span>)</span> <a href="lang1.html#K17">[*]</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:45_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K2" target="_self">fam_class</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font>
 <b>by </b><i>, <a class="txt" href="taxonom1.html#E3:45_1"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:45"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="taxonom1.html#K4" target="_self">fam_class_metr</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="taxonom1.html#K2" target="_self">fam_class</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">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">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K2" target="_self">fam_class</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font>
 ;<br/><b>consider </b><font color="Maroon">a</font> being  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> , <font color="Maroon">R</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E3:45_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <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">M</font>,<font color="Maroon">a</font></span>)</span> <a href="lang1.html#K17">[*]</a>  &amp;  <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">R1</font> = <font color="Maroon">R</font> as   <a href="alg_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">M</font> ;<br/><a NAME="E5:45_2"/>
<font color="Maroon">R1</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">M</font>,<font color="Maroon">a</font></span>)</span> <a href="lang1.html#K17">[*]</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:45_2"/>
<font color="Maroon">z</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>, </i>;<br/></div><b>end;</b></div>
<a NAME="E4:45"/><b>then </b>
 <a href="taxonom1.html#K2" target="_self">fam_class</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="taxonom1.html#K4" target="_self">fam_class_metr</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:45"/>
 <a href="taxonom1.html#K4" target="_self">fam_class_metr</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="taxonom1.html#K2" target="_self">fam_class</a> the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">M</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
