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

<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="E1:36"/><i><font color="Green">E24</font></i>: 
<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>and </b><br/><a NAME="E2:36"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>and </b><br/><a NAME="E3:36"/><i><font color="Green">E26</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#K2">-infty</a> 
 ;<br/>

<a NAME="E4:36"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1"/><i><font color="Green">E28</font></i>: 
(  not <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  or  not  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 </span><b>by </b><i><a class="txt" href="extreal1.html#E1:36_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_1"><b>suppose </b></a><a NAME="E1:36_1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:36_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="extreal1.html#E1:36"><i><font color="Green">E24</font></i></a>, <a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1_2"><b>suppose </b></a><a NAME="E1:36_1_1_1_2"/>
<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"><b>hence </b><a NAME="E2:36_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="extreal1.html#E1:36"><i><font color="Green">E24</font></i></a>, <a class="ref" href="supinf_1.html#T21">SUPINF_1:21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:36_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>4</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="extreal1.html#E2:36"><i><font color="Green">E25</font></i></a>, <a class="txt" href="extreal1.html#E3:36"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_2_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="36_2_1_1"><b>suppose </b></a><a NAME="E1:36_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> 
 ;<br/><div class="add"><a NAME="E2:36_2_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E2:36"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_2_1_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+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#K1">+infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:36_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:36_2_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E2:36"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:36_2_1_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:36_2_1_1"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:36_2_1_1_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#K1">+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="extreal1.html#E4:36"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><a NAME="E3:36_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="extreal1.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E4:36_2_1_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:36_2_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:36_2_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_2"><b>suppose </b></a><a NAME="E1:36_2_1_2"/><i><font color="Green">E28</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#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:36_2_1_2"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="txt" href="extreal1.html#E4:36"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</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="extreal1.html#E4:36"><i><font color="Green">E27</font></i></a>, <a class="txt" href="extreal1.html#E1:36_2_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><a NAME="E4:36_2_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="extreal1.html#T13" target="_self">Th13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_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="E1:36_2_1_2_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+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#K1">+infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:36_2_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:36_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E2:36"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:36_2_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E4:36_2_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:36_2_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#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:36_2_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="txt" href="extreal1.html#E4:36"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:36_2_1_2_1_1_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="extreal1.html#E1:36"><i><font color="Green">E24</font></i></a>, <a class="txt" href="extreal1.html#E2:36_2_1_2"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E5:36_2_1_2_1_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:36"><i><font color="Green">E24</font></i></a>, <a class="txt" href="extreal1.html#E4:36_2_1_2_1_1_2"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E6:36_2_1_2_1_1_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="extreal1.html#E4:36_2_1_2_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E7:36_2_1_2_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E2:36"><i><font color="Green">E25</font></i></a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><a NAME="E8:36_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="extreal1.html#E5:36_2_1_2_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/><b>thus </b><a NAME="E9:36_2_1_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E6:36_2_1_2_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="extreal1.html#E8:36_2_1_2_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:36_2_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:36"/>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div>
