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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:37"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">f</font> is <a href="metric_1.html#V4">symmetric</a> &amp; <font color="Maroon">f</font> is <a href="taxonom1.html#V1" target="_self">nonnegative</a> )
 ;<br/>

<a NAME="E2:37"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">rn</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finset_1.html#T26">FINSET_1:26</a>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A1</font> =  <a href="pre_circ.html#K1">max</a> <font color="Maroon">rn</font> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">A1</font> is <a href="xxreal_0.html#V3">negative</a>
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X</font>;<br/><a NAME="E3:37_1"/>
<font color="Maroon">f</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">A1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:37_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="taxonom1.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A1'</font> = <font color="Maroon">A1</font> 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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">X</font> ;<br/><a NAME="E3:37_2"/>
<font color="Maroon">f</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">A1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:37_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K1" target="_self">low_toler</a> <font color="Maroon">f</font>,<font color="Maroon">A1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E8:37"/><b>then </b>
 <a href="taxonom1.html#K1" target="_self">low_toler</a> <font color="Maroon">f</font>,<font color="Maroon">A1'</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">R</font> = <span class="p1">(<span class="default"><a href="taxonom1.html#K1" target="_self">low_toler</a> <font color="Maroon">f</font>,<font color="Maroon">A1'</font></span>)</span> <a href="lang1.html#K17">[*]</a>  as   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> <b>by </b><i>, </i>;<br/>
<a NAME="E10:37"/>
 <a href="eqrel_1.html#K9">Class</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K2" target="_self">fam_class</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E11:37"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">X</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K2" target="_self">fam_class</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:37"><i><font color="Green">E29</font></i></a></i>;<br/>


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