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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E32">A1</font></i>: 
(  <a href="eqrel_1.html#K10" title="EQREL_1:func.10">SmallestPartition</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="taxonom1.html#K2" title="TAXONOM1:func.2">fam_class</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> is <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a> &amp; <font color="Maroon" title="c2">f</font> is <a href="taxonom1.html#V1" title="TAXONOM1:attr.1">nonnegative</a> )
 ;<br/>

<a NAME="E2:39"/><i><font color="Green" title="E33">A2</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">X</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#K2" title="TAXONOM1:func.2">fam_class</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="taxonom1.html#E1:39"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="taxonom1.html#T30" target="_self" title="TAXONOM1:th.30">Th30</a></i>;<br/>
<a NAME="E3:39"/>
 <a href="taxonom1.html#K2" title="TAXONOM1:func.2">fam_class</a> <font color="Maroon" title="c2">f</font> is    <a href="taxonom1.html#M1" title="TAXONOM1:mode.1">Classification</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="taxonom1.html#T31" target="_self" title="TAXONOM1:th.31">Th31</a></i>;<br/>
<b>hence </b><a NAME="E4:39"/>
 <a href="taxonom1.html#K2" title="TAXONOM1:func.2">fam_class</a> <font color="Maroon" title="c2">f</font> is    <a href="taxonom1.html#M2" title="TAXONOM1:mode.2">Strong_Classification</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:39"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="taxonom1.html#E2:39"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="taxonom1.html#D2" target="_self" title="TAXONOM1:def.2">Def2</a></i>;<br/>


</div>
