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<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="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</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>let </b><font color="Maroon" title="c3">a</font> be  <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/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E24">A1</font></i>: 
(  <a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">a</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">f</font> is <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:28_1"/><i><font color="Green" title="E25">A2</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">a</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c5">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="taxonom1.html#E1:28_1"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/><a NAME="E3:28_1"/>
<font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:28_1"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="taxonom1.html#D3" target="_self" title="TAXONOM1:def.3">Def3</a></i>;<br/><a NAME="E4:28_1"/><b>then </b>
<font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c7">y1</font>,<font color="Maroon" title="c6">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:28"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="metric_1.html#D5" title="METRIC_1:def.5">METRIC_1:def 5</a></i>;<br/><b>hence </b><a NAME="E5:28_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="ref" href="taxonom1.html#D3" target="_self" title="TAXONOM1:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:28"/><b>then </b>
 <a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">a</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:28"/>
<span class="p1">(<span class="default"><a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:28"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="taxonom1.html#T9" target="_self" title="TAXONOM1:th.9">Th9</a></i>;<br/>


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