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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="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>let </b><font color="Maroon" title="c3">z</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="c4">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="E1:34"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c4">A</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c3">z</font> )
 ;<br/>

<a NAME="E2:34"/><i><font color="Green" title="E30">A2</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="c4">A</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <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="c4">A</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a> 
 
<b>by </b><i><a class="ref" href="lang1.html#T18" title="LANG1:th.18">LANG1:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:34_1"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="c4">A</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a> 
 ;<br/><b>consider </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:34_1"/><i><font color="Green" title="E32">A4</font></i>: 
( <font color="Maroon" title="c6">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="c7">y</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">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:34_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">x'</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">y'</font> = <font color="Maroon" title="c7">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="taxonom1.html#E3:34_1"><i><font color="Green" title="E32">A4</font></i></a></i>;<br/><a NAME="E5:34_1"/>
<font color="Maroon" title="c2">f</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c8">x'</font>,<font color="Maroon" title="c9">y'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:34"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="taxonom1.html#T26" target="_self" title="TAXONOM1:th.26">Th26</a></i>;<br/><b>hence </b><a NAME="E6:34_1"/>
<font color="Maroon" title="c5">p</font> <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="c4">A</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E3:34_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="taxonom1.html#D3" target="_self" title="TAXONOM1:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:34"/><b>then </b>
<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="c4">A</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">A</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:34"/>
<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="c4">A</font></span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a>  <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="taxonom1.html#K1" title="TAXONOM1:func.1">low_toler</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E2:34"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


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