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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">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c2">f</font> is <a href="taxonom1.html#V1" title="TAXONOM1:attr.1">nonnegative</a> &amp; <font color="Maroon" title="c2">f</font> is <a href="metric_1.html#V2" title="METRIC_1:attr.2">Reflexive</a> &amp; <font color="Maroon" title="c2">f</font> is <a href="metric_1.html#V3" title="METRIC_1:attr.3">discerning</a> )
 ;<br/>

<b>assume </b><a NAME="E2:26"/><i><font color="Green" title="E23">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c4">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="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:26"/>
<font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">y1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="taxonom1.html#E2:26"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="taxonom1.html#D3" target="_self" title="TAXONOM1:def.3">Def3</a></i>;<br/>
<a NAME="E5:26"/><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="c5">x1</font>,<font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="taxonom1.html#E1:26"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="taxonom1.html#D4" target="_self" title="TAXONOM1:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E6:26"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:26"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="metric_1.html#D4" title="METRIC_1:def.4">METRIC_1:def 4</a></i>;<br/>


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