<?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="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>assume </b><a NAME="E1:30"/><i><font color="Green" title="E26">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/>

<a NAME="E2:30"/><i><font color="Green" title="E27">A2</font></i>: 
 <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>  <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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:30_1"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c3">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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="lang1.html#K17" title="LANG1:func.17">[*]</a> 
 ;<br/><b>consider </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:30_1"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c3">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="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:30_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:30_1"/>
 <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>  <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:30_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="taxonom1.html#E3:30_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="rewrite1.html#T21" title="REWRITE1:th.21">REWRITE1:21</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">r</font> being    <a href="rewrite1.html#M1" title="REWRITE1:mode.1">RedSequence</a> of  <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> <b> such that </b><br/><a NAME="E6:30_1"/><i><font color="Green" title="E30">A5</font></i>: 
( <font color="Maroon" title="c6">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c6">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D3" title="REWRITE1:def.3">REWRITE1:def 3</a></i>;<br/><a NAME="E7:30_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">r</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></i>;<br/><a NAME="E8:30_1"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">r</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E9:30_1"/><b>then </b><i><font color="Green" title="E31">A6</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">r</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">r</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:30_1_1"/><i><font color="Green" title="E32">A7</font></i>: 
( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">r</font> &amp; <font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">r</font> )
 ;<br/><a NAME="E2:30_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></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> 
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:30_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="rewrite1.html#D2" title="REWRITE1:def.2">REWRITE1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:30_1_1"/>
<font color="Maroon" title="c6">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:30"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="taxonom1.html#T20" target="_self" title="TAXONOM1:th.20">Th20</a></i>;<br/></div><b>end;</b></div><a NAME="E11:30_1"/><b>then </b><i><font color="Green" title="E32">A8</font></i>: 
<font color="Maroon" title="c6">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="taxonom1.html#E9:30_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="taxonom1.html#T2" target="_self" title="TAXONOM1:th.2">Th2</a></i>;<br/><a NAME="E12:30_1"/>
<font color="Maroon" title="c4">x</font> is    <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#E1:30_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="taxonom1.html#E3:30_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E13:30_1"/>
<font color="Maroon" title="c3">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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:30"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="taxonom1.html#E3:30_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="taxonom1.html#E6:30_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="taxonom1.html#E11:30_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="taxonom1.html#T21" target="_self" title="TAXONOM1:th.21">Th21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:30"/><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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<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="E5:30"/>
<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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="taxonom1.html#E2:30"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
