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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:29"/><i><font color="Green">E20</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 <b>and </b><br/><a NAME="E2:29"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>thus </b><a NAME="E3:29"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/>

<a NAME="E2:29_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="txt" href="extreal2.html#E1:29"><i><font color="Green">E20</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:29_1"/>
contradiction
 <b>by </b><i><a class="txt" href="extreal2.html#E1:29"><i><font color="Green">E20</font></i></a>, <a class="txt" href="extreal2.html#E2:29"><i><font color="Green">E21</font></i></a>, <a class="txt" href="extreal2.html#E1:29_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/>


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

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1"><b>suppose </b></a><a NAME="E1:29_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:29_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E1:29"><i><font color="Green">E20</font></i></a>, <a class="ref" href="supinf_2.html#T6">SUPINF_2:6</a></i>;<br/><b>hence </b><a NAME="E3:29_2_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2"><b>suppose </b></a><a NAME="E1:29_2_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:29_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/><a NAME="E3:29_2_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E1:29_2_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:29_2_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E2:29"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:29_2_2"/><b>then </b><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="txt" href="extreal2.html#E1:29"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal2.html#T23" target="_self">Th23</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_2_2_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_2_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:29_2_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T5">XXREAL_0:5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_2_2_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="extreal2.html#E5:29_2_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_2_2_1_1_2_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:29_2_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:29_2_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E1:29"><i><font color="Green">E20</font></i></a>, <a class="ref" href="supinf_2.html#T7">SUPINF_2:7</a></i>;<br/><b>hence </b><a NAME="E3:29_2_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:29_2_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="extreal2.html#E5:29_2_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><a NAME="E3:29_2_2_1_1_2_1_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/><a NAME="E4:29_2_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E2:29"><i><font color="Green">E21</font></i></a>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="extreal2.html#E1:29_2_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><a NAME="E6:29_2_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:29_2_2_1_1_2_1_1_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="extreal2.html#E2:29"><i><font color="Green">E21</font></i></a>, <a class="txt" href="extreal2.html#E3:29_2_2_1_1_2_1_1_2"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E9:29_2_2_1_1_2_1_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="extreal2.html#E3:29_2_2_1_1_2_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="extreal2.html#E8:29_2_2_1_1_2_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E10:29_2_2_1_1_2_1_1_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>;<br/><b>hence </b><a NAME="E11:29_2_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="extreal2.html#E9:29_2_2_1_1_2_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:29_2_2_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:29_2_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
