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

<b>let </b><font color="Maroon" title="c2">X</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E36">A1</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c2">X</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c3">Z</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>  :  ex <font color="Olive" title="b2">U</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">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="Olive" title="b2">U</font> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c4">z</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>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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>  :  ex <font color="Olive" title="b2">U</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">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="Olive" title="b2">U</font> ) </span>}</span> ;<br/>
<a NAME="E3:46"/>
 <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>  :  ex <font color="Olive" title="b2">U</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">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="Olive" title="b2">U</font> ) </span>}</span>  <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="topgen_3.html#K3" title="TOPGEN_3:func.3">continuum</a> 
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:46"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="topgen_3.html#T31" target="_self" title="TOPGEN_3:th.31">Th31</a></i>;<br/>
<a NAME="E4:46"/><b>then </b><i><font color="Green" title="E37">A2</font></i>: 
<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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>  :  ex <font color="Olive" title="b2">U</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">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="Olive" title="b2">U</font> ) </span>}</span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K13" title="ARYTM_3:func.13">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T4" title="CARD_2:th.4">CARD_2:4</a></i>;<br/>
<a NAME="E5:46"/><b>then </b><i><font color="Green" title="E38">A3</font></i>: 
( <font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; not <font color="Maroon" title="c4">z</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>  :  ex <font color="Olive" title="b2">U</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">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="Olive" title="b2">U</font> ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c4">z</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#E4:46"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:46"/>
<font color="Maroon" title="c5">z</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">z</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">q</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="E9:46"/><i><font color="Green" title="E39">A4</font></i>: 
( <font color="Maroon" title="c5">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">q</font> &amp; <font color="Maroon" title="c6">q</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">z</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 <b>by </b><i><a class="ref" href="rat_1.html#T22" title="RAT_1:th.22">RAT_1:22</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">z</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c6">q</font>
;<br/>

<b>thus </b><a NAME="E10:46"/>
<font color="Maroon" title="c5">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">q</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E9:46"><i><font color="Green" title="E39">A4</font></i></a></i>;<br/>

<a NAME="E11:46"/>
<font color="Maroon" title="c5">z</font> <a href="topgen_3.html#R1" title="TOPGEN_3:pred.1">is_local_minimum_of</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c6">q</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:46_1"/>
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c6">q</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 <b>by </b><i><a class="txt" href="topgen_3.html#E9:46"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="rcomp_2.html#T3" title="RCOMP_2:th.3">RCOMP_2:3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="topgen_3.html#D3" target="_self" title="TOPGEN_3:def.3">TOPGEN_3:def 3</a><br/>

<b>take </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c5">z</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/>

<a NAME="E2:46_1"/>
<font color="Maroon" title="c5">z</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font>
 
;<br/>
<b>hence </b><a NAME="E3:46_1"/>
<font color="Maroon" title="c7">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">z</font>
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>

<b>assume </b><a NAME="E4:46_1"/>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c5">z</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c6">q</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:46_1"/><i><font color="Green" title="E40">A5</font></i>: 
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c5">z</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c6">q</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> )
 <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="c9">a</font> = <font color="Maroon" title="c8">a</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#E6:46_1"><i><font color="Green" title="E40">A5</font></i></a></i>;<br/>
<a NAME="E8:46_1"/>
( <font color="Maroon" title="c9">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c5">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">a</font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E6:46_1"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="rcomp_2.html#T3" title="RCOMP_2:th.3">RCOMP_2:3</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="E9:46_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:46"/>
not <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c6">q</font></span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E5:46"><i><font color="Green" title="E38">A3</font></i></a></i>;<br/>


</div>
