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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V3" target="_self">bounded</a>
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green">E22</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>thus </b><a NAME="E3:19"/>
( ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> ) implies  <a href="seq_4.html#K5" target="_self">lower_bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="seq_4.html#K4" target="_self">upper_bound</a> <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><a NAME="E1:19_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:19_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:19_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:19_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V2" target="_self">bounded_below</a>
 <b>by </b><i><a class="txt" href="seq_4.html#E1:19"><i><font color="Green">E21</font></i></a>, <a class="ref" href="seq_4.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E3:19_1_1"/><b>then </b>
 <a href="seq_4.html#K5" target="_self">lower_bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="seq_4.html#E2:19_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="seq_4.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:19_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="seq_4.html#K5" target="_self">lower_bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="seq_4.html#E1:19_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:19_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V1" target="_self">bounded_above</a>
 <b>by </b><i><a class="txt" href="seq_4.html#E1:19"><i><font color="Green">E21</font></i></a>, <a class="ref" href="seq_4.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E6:19_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_4.html#K4" target="_self">upper_bound</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="seq_4.html#E1:19_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="seq_4.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E7:19_1_1"/>
 <a href="seq_4.html#K5" target="_self">lower_bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="seq_4.html#K4" target="_self">upper_bound</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="seq_4.html#E4:19_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:19_1_2"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V2" target="_self">bounded_below</a>
 <b>by </b><i><a class="txt" href="seq_4.html#E1:19"><i><font color="Green">E21</font></i></a>, <a class="ref" href="seq_4.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E3:19_1_2"/><b>then </b>
 <a href="seq_4.html#K5" target="_self">lower_bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="seq_4.html#E1:19_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="seq_4.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E4:19_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="seq_4.html#K5" target="_self">lower_bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="seq_4.html#E1:19_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:19_1_2"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V1" target="_self">bounded_above</a>
 <b>by </b><i><a class="txt" href="seq_4.html#E1:19"><i><font color="Green">E21</font></i></a>, <a class="ref" href="seq_4.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E6:19_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_4.html#K4" target="_self">upper_bound</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="seq_4.html#E2:19_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="seq_4.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E7:19_1_2"/>
 <a href="seq_4.html#K5" target="_self">lower_bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="seq_4.html#K4" target="_self">upper_bound</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="seq_4.html#E4:19_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:19_1"/>
 <a href="seq_4.html#K5" target="_self">lower_bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="seq_4.html#K4" target="_self">upper_bound</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="seq_4.html#E3:19_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="seq_4.html#E4:19_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E4:19"/><i><font color="Green">E23</font></i>: 
 <a href="seq_4.html#K5" target="_self">lower_bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="seq_4.html#K4" target="_self">upper_bound</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:19"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></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="E7:19"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="seq_4.html#E2:19"><i><font color="Green">E22</font></i></a>, <a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_2_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:19_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is   <a href="real_1.html#NM1">Real</a>
 ;<br/><b>hence </b><a NAME="E3:19_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="seq_4.html#E5:19"><i><font color="Green">E24</font></i></a>, <a class="txt" href="seq_4.html#E7:19"><i><font color="Green">E25</font></i></a>, <a class="txt" href="seq_4.html#E1:19_2_1"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:19_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> iff <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="seq_4.html#E7:19"><i><font color="Green">E25</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:19"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1" target="_self">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="seq_4.html#K1" target="_self">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:19"/>
contradiction
 <b>by </b><i><a class="txt" href="seq_4.html#E4:19"><i><font color="Green">E23</font></i></a>, <a class="ref" href="seq_4.html#T23" target="_self">Th23</a></i>;<br/>


</div>
