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

<b>let </b><font color="Maroon">U</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">x</font> <a href="topgen_3.html#R1" target="_self">is_local_minimum_of</a> <font color="Maroon">U</font> </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">x</font>, <font color="Olive">y</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">y</font>,<font color="Olive">x</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">x</font> &amp; <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">y</font>,<font color="Olive">x</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">U</font> );<br/>
<div><i><font color="Green">E37</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">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:28_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">x</font> <a href="topgen_3.html#R1" target="_self">is_local_minimum_of</a> <font color="Maroon">U</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="topgen_3.html#R1" target="_self">is_local_minimum_of</a> <font color="Maroon">U</font> )
 ;<br/><a NAME="E4:28_1"/>
 ex <font color="Olive">y</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">x</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">U</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:28_1"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Olive">b</font>] )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:28"/><i><font color="Green">E39</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">x</font> <a href="topgen_3.html#R1" target="_self">is_local_minimum_of</a> <font color="Maroon">U</font> </span>}</span>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K1">REAL</a>  )
 <b>and </b><br/><a NAME="E4:28"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">x</font> <a href="topgen_3.html#R1" target="_self">is_local_minimum_of</a> <font color="Maroon">U</font> </span>}</span>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<a NAME="E5:28"/><i><font color="Green">E46</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> is <a href="taxonom2.html#V4">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">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="taxonom2.html#D5">TAXONOM2:def 5</a><br/><b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:28_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:28_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a'</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>assume </b><a NAME="E4:28_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:28_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">xa</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">ya</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E8:28_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xa</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">ya</font>,<font color="Maroon">xa</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <font color="Maroon">ya</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">xa</font> &amp; <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">ya</font>,<font color="Maroon">xa</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">U</font> )
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">xb</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">yb</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E10:28_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xb</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">yb</font>,<font color="Maroon">xb</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <font color="Maroon">yb</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">xb</font> &amp; <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">yb</font>,<font color="Maroon">xb</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">U</font> )
 <b>by </b><i>, , </i>;<br/>
<b>assume </b><a NAME="E11:28_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">b</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:28_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E15:28_2"/>
(  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">xa</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="topgen_3.html#R1" target="_self">is_local_minimum_of</a> <font color="Maroon">U</font> ) &amp;  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">xb</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="topgen_3.html#R1" target="_self">is_local_minimum_of</a> <font color="Maroon">U</font> ) )
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E16:28_2"/><b>then </b><i><font color="Green">E62</font></i>: 
( <font color="Maroon">xb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; <font color="Maroon">xa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E17:28_2"/>
( <font color="Maroon">yb</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">xa</font> &amp; <font color="Maroon">ya</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">xb</font> )
 
<b>by </b><i>, , , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E18:28_2"/><b>then </b>
( ( <font color="Maroon">yb</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">xa</font> &amp; <font color="Maroon">xa</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">xb</font> ) or ( <font color="Maroon">xa</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">xb</font> &amp; <font color="Maroon">xb</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">ya</font> ) )
 
<b>by </b><i>, , , , <a class="ref" href="topgen_3.html#D1" target="_self">Def1</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:28_2"/><b>then </b>
( <font color="Maroon">xa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> or <font color="Maroon">xb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<b>hence </b><a NAME="E20:28_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">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">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:28_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/><b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:28_3"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">y</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E5:28_3"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">U</font> )
 <b>by </b><i>, , </i>;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>take </b><font color="Maroon">x</font> = <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E6:28_3"/>
( <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E7:28"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> is <a href="card_4.html#V1">countable</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:28"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0
 
<b>by </b><i><a class="ref" href="card_4.html#D1">CARD_4:def 1</a></i>;<br/>
<a NAME="E9:28"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">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">a'</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b'</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>set </b><font color="Maroon">a</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a'</font>;<br/>
<b>set </b><font color="Maroon">b</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:28_4"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:28_4"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>consider </b><font color="Maroon">xa</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">ya</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:28_4"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xa</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">ya</font>,<font color="Maroon">xa</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <font color="Maroon">ya</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">xa</font> &amp; <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">ya</font>,<font color="Maroon">xa</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">U</font> )
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">xb</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">yb</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E6:28_4"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xb</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">yb</font>,<font color="Maroon">xb</font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <font color="Maroon">yb</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">xb</font> &amp; <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">yb</font>,<font color="Maroon">xb</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">U</font> )
 <b>by </b><i>, , </i>;<br/>
<b>assume </b><a NAME="E7:28_4"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font> )
 ;<br/>

<b>consider </b><font color="Maroon">c</font> being  <a href="rat_1.html#V1">rational</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E9:28_4"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">ya</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">xa</font> )
 <b>by </b><i>, <a class="ref" href="rat_1.html#T22">RAT_1:22</a></i>;<br/>
<a NAME="E10:28_4"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a'</font>
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E11:28_4"/>
(  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">xa</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="topgen_3.html#R1" target="_self">is_local_minimum_of</a> <font color="Maroon">U</font> ) &amp;  ex <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">xb</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="topgen_3.html#R1" target="_self">is_local_minimum_of</a> <font color="Maroon">U</font> ) )
 
<b>by </b><i>, , , , <a class="ref" href="topgen_3.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E12:28_4"/><b>then </b><i><font color="Green">E132</font></i>: 
( <font color="Maroon">xb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; <font color="Maroon">xa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E13:28_4"/>
( <font color="Maroon">yb</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">xa</font> &amp; <font color="Maroon">ya</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">xb</font> )
 
<b>by </b><i>, <a class="ref" href="topgen_3.html#D1" target="_self">Def1</a>, , , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E14:28_4"/><b>then </b>
( ( <font color="Maroon">yb</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">xa</font> &amp; <font color="Maroon">xa</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">xb</font> ) or ( <font color="Maroon">xa</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">xb</font> &amp; <font color="Maroon">xb</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">ya</font> ) )
 
<b>by </b><i>, <a class="ref" href="topgen_3.html#D1" target="_self">Def1</a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:28_4"/><b>then </b>
( <font color="Maroon">xa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b'</font> or <font color="Maroon">xb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a'</font> )
 
<b>by </b><i>, <a class="ref" href="topgen_3.html#D1" target="_self">Def1</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<b>hence </b><a NAME="E16:28_4"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="topgen_3.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:28"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">x</font> <a href="topgen_3.html#R1" target="_self">is_local_minimum_of</a> <font color="Maroon">U</font> </span>}</span>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T26">CARD_1:26</a></i>;<br/>
<b>hence </b><a NAME="E11:28"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Olive">x</font> <a href="topgen_3.html#R1" target="_self">is_local_minimum_of</a> <font color="Maroon">U</font> </span>}</span>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K3">alef</a> 0
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="card_4.html#D1">CARD_4:def 1</a><br/>


</div>
