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

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:46"/><i><font color="Green">E36</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>2</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="topgen_3.html#K3" target="_self">continuum</a> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="topgen_3.html#R1" target="_self">is_local_minimum_of</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of <a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="topgen_3.html#R1" target="_self">is_local_minimum_of</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> ;<br/>
<a NAME="E3:46"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="topgen_3.html#R1" target="_self">is_local_minimum_of</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span>  <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="topgen_3.html#K3" target="_self">continuum</a> 
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:46"><i><font color="Green">E36</font></i></a>, <a class="ref" href="topgen_3.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E4:46"/><b>then </b>
<a href="numbers.html#K1">REAL</a>  <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="topgen_3.html#R1" target="_self">is_local_minimum_of</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="arytm_3.html#K13">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T4">CARD_2:4</a></i>;<br/>
<a NAME="E5:46"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="topgen_3.html#R1" target="_self">is_local_minimum_of</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E7:46"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></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="E9:46"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 )
 <b>by </b><i><a class="ref" href="rat_1.html#T22">RAT_1:22</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E10:46"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E9:46"><i><font color="Green">E38</font></i></a></i>;<br/>

<a NAME="E11:46"/>
<font color="Maroon">c<sub>5</sub></font> <a href="topgen_3.html#R1" target="_self">is_local_minimum_of</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_2.html#K1">.[</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">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_2.html#K1">.[</a></span>
 <b>by </b><i><a class="txt" href="topgen_3.html#E9:46"><i><font color="Green">E38</font></i></a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="topgen_3.html#D3" target="_self">TOPGEN_3:def 3</a><br/>

<b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> 1;<br/>

<a NAME="E2:46_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xcmplx_0.html#K6">-</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<b>hence </b><a NAME="E3:46_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>

<b>assume </b><a NAME="E4:46_1"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_2.html#K1">.[</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:46_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="rcomp_1.html#K2">.[</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> )
 <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<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="topgen_3.html#E6:46_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E8:46_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E6:46_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="rcomp_2.html#T3">RCOMP_2:3</a>, <a class="ref" href="rcomp_1.html#T47">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">[.</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="rcomp_2.html#K1">.[</a></span> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E5:46"><i><font color="Green">E37</font></i></a></i>;<br/>


</div>
