<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c2">U</font> be   <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>set </b><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : <font color="Olive" title="b1">x</font> <a href="topgen_3.html#R1" title="TOPGEN_3:pred.1">is_local_minimum_of</a> <font color="Maroon" title="c2">U</font> </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">U</font> );<br/>
<div><i><font color="Green" title="E23">A1</font></i>: <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">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:28_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : <font color="Olive" title="b1">x</font> <a href="topgen_3.html#R1" title="TOPGEN_3:pred.1">is_local_minimum_of</a> <font color="Maroon" title="c2">U</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green" title="E24">A2</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c5">x</font> <a href="topgen_3.html#R1" title="TOPGEN_3:pred.1">is_local_minimum_of</a> <font color="Maroon" title="c2">U</font> )
 ;<br/><a NAME="E4:28_1"/>
 ex <font color="Olive" title="b1">y</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">x</font> &amp; <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">U</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:28_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="topgen_3.html#D3" target="_self" title="TOPGEN_3:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E5:28_1"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">a</font>,<font color="Olive" title="b1">b</font>] )
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:28_1"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:28"/><i><font color="Green" title="E24">A3</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : <font color="Olive" title="b1">x</font> <a href="topgen_3.html#R1" title="TOPGEN_3:pred.1">is_local_minimum_of</a> <font color="Maroon" title="c2">U</font> </span>}</span>  &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 <b>and </b><br/><a NAME="E4:28"/><i><font color="Green" title="E25">A4</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : <font color="Olive" title="b2">x</font> <a href="topgen_3.html#R1" title="TOPGEN_3:pred.1">is_local_minimum_of</a> <font color="Maroon" title="c2">U</font> </span>}</span>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="topgen_3.html#E1:28"><i><font color="Green" title="E23">A1</font></i></a>);<br/></i>
<a NAME="E5:28"/><i><font color="Green" title="E26">A5</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> is <a href="taxonom2.html#V4" title="TAXONOM2:attr.4">mutually-disjoint</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="taxonom2.html#D5" title="TAXONOM2:def.5">TAXONOM2:def 5</a><br/><b>let </b><font color="Maroon" title="c6">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:28_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">a'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:28_2"/><i><font color="Green" title="E27">A6</font></i>: 
( <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a'</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>assume </b><a NAME="E4:28_2"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">b'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:28_2"/><i><font color="Green" title="E28">A7</font></i>: 
( <font color="Maroon" title="c8">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b'</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">xa</font>, <font color="Maroon" title="c10">ya</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E8:28_2"/><i><font color="Green" title="E29">A8</font></i>: 
( <font color="Maroon" title="c7">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">xa</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c10">ya</font>,<font color="Maroon" title="c9">xa</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Maroon" title="c10">ya</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">xa</font> &amp; <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c10">ya</font>,<font color="Maroon" title="c9">xa</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">U</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:28"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="topgen_3.html#E4:28"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="topgen_3.html#E3:28_2"><i><font color="Green" title="E27">A6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">xb</font>, <font color="Maroon" title="c12">yb</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E10:28_2"/><i><font color="Green" title="E30">A9</font></i>: 
( <font color="Maroon" title="c8">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">xb</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c12">yb</font>,<font color="Maroon" title="c11">xb</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Maroon" title="c12">yb</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">xb</font> &amp; <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c12">yb</font>,<font color="Maroon" title="c11">xb</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">U</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:28"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="topgen_3.html#E4:28"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="topgen_3.html#E6:28_2"><i><font color="Green" title="E28">A7</font></i></a></i>;<br/>
<b>assume </b><a NAME="E11:28_2"/><i><font color="Green" title="E31">A10</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c5">a</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">b</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:28_2"/><i><font color="Green" title="E32">A11</font></i>: 
( <font color="Maroon" title="c13">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c13">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">c</font> = <font color="Maroon" title="c13">c</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="topgen_3.html#E8:28_2"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="topgen_3.html#E13:28_2"><i><font color="Green" title="E32">A11</font></i></a></i>;<br/>
<a NAME="E15:28_2"/>
(  ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c9">xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="topgen_3.html#R1" title="TOPGEN_3:pred.1">is_local_minimum_of</a> <font color="Maroon" title="c2">U</font> ) &amp;  ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c11">xb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="topgen_3.html#R1" title="TOPGEN_3:pred.1">is_local_minimum_of</a> <font color="Maroon" title="c2">U</font> ) )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E3:28"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="topgen_3.html#E3:28_2"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="topgen_3.html#E6:28_2"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="topgen_3.html#E8:28_2"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="topgen_3.html#E10:28_2"><i><font color="Green" title="E30">A9</font></i></a></i>;<br/>
<a NAME="E16:28_2"/><b>then </b><i><font color="Green" title="E33">A12</font></i>: 
( <font color="Maroon" title="c11">xb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font> &amp; <font color="Maroon" title="c9">xa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font> )
 
<b>by </b><i><a class="ref" href="topgen_3.html#D3" target="_self" title="TOPGEN_3:def.3">Def3</a></i>;<br/>
<a NAME="E17:28_2"/>
( <font color="Maroon" title="c12">yb</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">c</font> &amp; <font color="Maroon" title="c14">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">xa</font> &amp; <font color="Maroon" title="c10">ya</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">c</font> &amp; <font color="Maroon" title="c14">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">xb</font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E8:28_2"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="topgen_3.html#E10:28_2"><i><font color="Green" title="E30">A9</font></i></a>, <a class="txt" href="topgen_3.html#E13:28_2"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<a NAME="E18:28_2"/><b>then </b>
( ( <font color="Maroon" title="c12">yb</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">xa</font> &amp; <font color="Maroon" title="c9">xa</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">xb</font> ) or ( <font color="Maroon" title="c9">xa</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">xb</font> &amp; <font color="Maroon" title="c11">xb</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">ya</font> ) )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E3:28_2"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="topgen_3.html#E6:28_2"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="topgen_3.html#E8:28_2"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="topgen_3.html#E10:28_2"><i><font color="Green" title="E30">A9</font></i></a>, <a class="txt" href="topgen_3.html#E11:28_2"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:28_2"/><b>then </b>
( <font color="Maroon" title="c9">xa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">b</font> or <font color="Maroon" title="c11">xb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E8:28_2"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="topgen_3.html#E10:28_2"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<b>hence </b><a NAME="E20:28_2"/>
contradiction
 <b>by </b><i><a class="txt" href="topgen_3.html#E16:28_2"><i><font color="Green" title="E33">A12</font></i></a>, <a class="txt" href="topgen_3.html#E8:28_2"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="topgen_3.html#E10:28_2"><i><font color="Green" title="E30">A9</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:28_3"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:28_3"/><i><font color="Green" title="E27">A13</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E5:28_3"/><i><font color="Green" title="E28">A14</font></i>: 
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Maroon" title="c8">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">x</font> &amp; <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c7">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">U</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:28"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="topgen_3.html#E4:28"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="topgen_3.html#E3:28_3"><i><font color="Green" title="E27">A13</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c8">y</font>;<br/><b>take </b><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c7">x</font>;<br/><b>thus </b><a NAME="E6:28_3"/>
( <font color="Maroon" title="c9">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">x</font> &amp; <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E5:28_3"><i><font color="Green" title="E28">A14</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:28"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E5:28"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="topgen_3.html#T18" target="_self" title="TOPGEN_3:th.18">Th18</a></i>;<br/>
<a NAME="E8:28"/><b>then </b><i><font color="Green" title="E27">A15</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0
 
<b>by </b><i><a class="ref" href="card_4.html#D1" title="CARD_4:def.1">CARD_4:def 1</a></i>;<br/>
<a NAME="E9:28"/>
<font color="Maroon" title="c4">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a'</font>, <font color="Maroon" title="c6">b'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>set </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a'</font>;<br/>
<b>set </b><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b'</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:28_4"/><i><font color="Green" title="E28">A16</font></i>: 
<font color="Maroon" title="c5">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E2:28_4"/><i><font color="Green" title="E29">A17</font></i>: 
<font color="Maroon" title="c6">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">xa</font>, <font color="Maroon" title="c10">ya</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:28_4"/><i><font color="Green" title="E30">A18</font></i>: 
( <font color="Maroon" title="c5">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">xa</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c10">ya</font>,<font color="Maroon" title="c9">xa</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Maroon" title="c10">ya</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">xa</font> &amp; <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c10">ya</font>,<font color="Maroon" title="c9">xa</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">U</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:28"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="topgen_3.html#E4:28"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="topgen_3.html#E1:28_4"><i><font color="Green" title="E28">A16</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">xb</font>, <font color="Maroon" title="c12">yb</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E6:28_4"/><i><font color="Green" title="E31">A19</font></i>: 
( <font color="Maroon" title="c6">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">xb</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c12">yb</font>,<font color="Maroon" title="c11">xb</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Maroon" title="c12">yb</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">xb</font> &amp; <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c12">yb</font>,<font color="Maroon" title="c11">xb</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">U</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E3:28"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="topgen_3.html#E4:28"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="topgen_3.html#E2:28_4"><i><font color="Green" title="E29">A17</font></i></a></i>;<br/>
<b>assume </b><a NAME="E7:28_4"/><i><font color="Green" title="E32">A20</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b'</font> &amp; <font color="Maroon" title="c5">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b'</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c13">c</font> being  <a href="rat_1.html#V1" title="RAT_1:attr.1">rational</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E9:28_4"/><i><font color="Green" title="E33">A21</font></i>: 
( <font color="Maroon" title="c10">ya</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">c</font> &amp; <font color="Maroon" title="c13">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">xa</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E4:28_4"><i><font color="Green" title="E30">A18</font></i></a>, <a class="ref" href="rat_1.html#T22" title="RAT_1:th.22">RAT_1:22</a></i>;<br/>
<a NAME="E10:28_4"/><i><font color="Green" title="E34">A22</font></i>: 
<font color="Maroon" title="c13">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a'</font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E4:28_4"><i><font color="Green" title="E30">A18</font></i></a>, <a class="txt" href="topgen_3.html#E9:28_4"><i><font color="Green" title="E33">A21</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<a NAME="E11:28_4"/>
(  ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c9">xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="topgen_3.html#R1" title="TOPGEN_3:pred.1">is_local_minimum_of</a> <font color="Maroon" title="c2">U</font> ) &amp;  ex <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <br/>( <font color="Maroon" title="c11">xb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="topgen_3.html#R1" title="TOPGEN_3:pred.1">is_local_minimum_of</a> <font color="Maroon" title="c2">U</font> ) )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E3:28"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="topgen_3.html#E1:28_4"><i><font color="Green" title="E28">A16</font></i></a>, <a class="txt" href="topgen_3.html#E2:28_4"><i><font color="Green" title="E29">A17</font></i></a>, <a class="txt" href="topgen_3.html#E4:28_4"><i><font color="Green" title="E30">A18</font></i></a>, <a class="txt" href="topgen_3.html#E6:28_4"><i><font color="Green" title="E31">A19</font></i></a></i>;<br/>
<a NAME="E12:28_4"/><b>then </b><i><font color="Green" title="E35">A23</font></i>: 
( <font color="Maroon" title="c11">xb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font> &amp; <font color="Maroon" title="c9">xa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font> )
 
<b>by </b><i><a class="ref" href="topgen_3.html#D3" target="_self" title="TOPGEN_3:def.3">Def3</a></i>;<br/>
<a NAME="E13:28_4"/>
( <font color="Maroon" title="c12">yb</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">c</font> &amp; <font color="Maroon" title="c13">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">xa</font> &amp; <font color="Maroon" title="c10">ya</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">c</font> &amp; <font color="Maroon" title="c13">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">xb</font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E4:28_4"><i><font color="Green" title="E30">A18</font></i></a>, <a class="txt" href="topgen_3.html#E6:28_4"><i><font color="Green" title="E31">A19</font></i></a>, <a class="txt" href="topgen_3.html#E7:28_4"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="topgen_3.html#E10:28_4"><i><font color="Green" title="E34">A22</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<a NAME="E14:28_4"/><b>then </b>
( ( <font color="Maroon" title="c12">yb</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">xa</font> &amp; <font color="Maroon" title="c9">xa</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">xb</font> ) or ( <font color="Maroon" title="c9">xa</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">xb</font> &amp; <font color="Maroon" title="c11">xb</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">ya</font> ) )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E4:28_4"><i><font color="Green" title="E30">A18</font></i></a>, <a class="txt" href="topgen_3.html#E6:28_4"><i><font color="Green" title="E31">A19</font></i></a>, <a class="txt" href="topgen_3.html#E7:28_4"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:28_4"/><b>then </b>
( <font color="Maroon" title="c9">xa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b'</font> or <font color="Maroon" title="c11">xb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a'</font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E4:28_4"><i><font color="Green" title="E30">A18</font></i></a>, <a class="txt" href="topgen_3.html#E6:28_4"><i><font color="Green" title="E31">A19</font></i></a>, <a class="ref" href="rcomp_1.html#T47" title="RCOMP_1:th.47">RCOMP_1:47</a></i>;<br/>
<b>hence </b><a NAME="E16:28_4"/>
contradiction
 <b>by </b><i><a class="txt" href="topgen_3.html#E12:28_4"><i><font color="Green" title="E35">A23</font></i></a>, <a class="txt" href="topgen_3.html#E4:28_4"><i><font color="Green" title="E30">A18</font></i></a>, <a class="txt" href="topgen_3.html#E6:28_4"><i><font color="Green" title="E31">A19</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:28"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : <font color="Olive" title="b1">x</font> <a href="topgen_3.html#R1" title="TOPGEN_3:pred.1">is_local_minimum_of</a> <font color="Maroon" title="c2">U</font> </span>}</span>  <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E3:28"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="card_1.html#T26" title="CARD_1:th.26">CARD_1:26</a></i>;<br/>
<b>hence </b><a NAME="E11:28"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : <font color="Olive" title="b1">x</font> <a href="topgen_3.html#R1" title="TOPGEN_3:pred.1">is_local_minimum_of</a> <font color="Maroon" title="c2">U</font> </span>}</span>  <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K3" title="CARD_1:func.3">alef</a> 0
 <b>by </b><i><a class="txt" href="topgen_3.html#E8:28"><i><font color="Green" title="E27">A15</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="card_4.html#D1" title="CARD_4:def.1">CARD_4:def 1</a><br/>


</div>
