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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</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="E1:30"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">f</font> is <a href="taxonom1.html#V1" target="_self">nonnegative</a> &amp; <font color="Maroon">f</font> is <a href="metric_1.html#V2">Reflexive</a> &amp; <font color="Maroon">f</font> is <a href="metric_1.html#V3">discerning</a> )
 ;<br/>

<a NAME="E2:30"/><i><font color="Green">E30</font></i>: 
 <a href="taxonom1.html#K1" target="_self">low_toler</a> <font color="Maroon">f</font>,0 <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" target="_self">low_toler</a> <font color="Maroon">f</font>,0</span>)</span> <a href="lang1.html#K17">[*]</a> 
 
<b>by </b><i><a class="ref" href="lang1.html#T18">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">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:30_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="taxonom1.html#K1" target="_self">low_toler</a> <font color="Maroon">f</font>,0</span>)</span> <a href="lang1.html#K17">[*]</a> 
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:30_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:30_1"/>
 <a href="taxonom1.html#K1" target="_self">low_toler</a> <font color="Maroon">f</font>,0 <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i>, <a class="txt" href="taxonom1.html#E1:30"><i><font color="Green">E29</font></i></a>, <a class="ref" href="rewrite1.html#T21">REWRITE1:21</a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being    <a href="rewrite1.html#M1">RedSequence</a> of  <a href="taxonom1.html#K1" target="_self">low_toler</a> <font color="Maroon">f</font>,0<b> such that </b><br/><a NAME="E6:30_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="rewrite1.html#D3">REWRITE1:def 3</a></i>;<br/><a NAME="E7:30_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#D2">REWRITE1:def 2</a></i>;<br/><a NAME="E8:30_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:30_1"/><b>then </b><i><font color="Green">E38</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">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">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:30_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> &amp; <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">r</font> )
 ;<br/><a NAME="E2:30_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></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>,0
 <b>by </b><i><a class="txt" href="taxonom1.html#E3:30_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="rewrite1.html#D2">REWRITE1:def 2</a></i>;<br/><b>hence </b><a NAME="E3:30_1_1"/>
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="taxonom1.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><a NAME="E11:30_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="taxonom1.html#E1:30_1"><i><font color="Green">E31</font></i></a>, </i>;<br/><a NAME="E12:30_1"/>
<font color="Maroon">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="taxonom1.html#E1:30"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E13:30_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="taxonom1.html#K1" target="_self">low_toler</a> <font color="Maroon">f</font>,0
 <b>by </b><i><a class="ref" href="taxonom1.html#T1" target="_self">Th1</a>, <a class="txt" href="taxonom1.html#E1:30"><i><font color="Green">E29</font></i></a>, <a class="txt" href="taxonom1.html#E2:30"><i><font color="Green">E30</font></i></a>, <a class="txt" href="taxonom1.html#E3:30_1"><i><font color="Green">E32</font></i></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" target="_self">low_toler</a> <font color="Maroon">f</font>,0</span>)</span> <a href="lang1.html#K17">[*]</a>  <a href="tarski.html#R1">c=</a>  <a href="taxonom1.html#K1" target="_self">low_toler</a> <font color="Maroon">f</font>,0
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:30"/>
<span class="p1">(<span class="default"><a href="taxonom1.html#K1" target="_self">low_toler</a> <font color="Maroon">f</font>,0</span>)</span> <a href="lang1.html#K17">[*]</a>  <a href="hidden.html#R1">=</a>  <a href="taxonom1.html#K1" target="_self">low_toler</a> <font color="Maroon">f</font>,0
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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