<?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/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1"/>
( ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>2</sub></font> ) or (  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or (  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <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>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</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; <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <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="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <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>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</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; <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <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="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) or (  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) or (  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <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>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) )
 </span><b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:17_1_1_1"/><i><font color="Green">E18</font></i>: 
( <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>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</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#K4">*</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><a NAME="E4:17_1_1_1"/><i><font color="Green">E19</font></i>: 
( <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>  &amp; <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>  )
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_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="E1:17_1_1_1_1_1"/>
( ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</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="Olive">b<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>2</sub></font> ) or (  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</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="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>  ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">c<sub>3</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="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>  ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</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; <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>  ) or (  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</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="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>  ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &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>  <a href="supinf_2.html#K1">0.</a>  ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &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>  <a href="supinf_2.html#K1">0.</a>  ) )
 </span><b>by </b><i><a class="txt" href="extreal1.html#E4:17_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</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="Olive">b<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/><div class="add"><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="E3:17_1_1_1_1_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</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>6</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/><a NAME="E4:17_1_1_1_1_1_1"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E3:17_1_1_1_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E3:17_1_1_1_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a></i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_1"><i><font color="Green">E21</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_2"/><i><font color="Green">E20</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</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="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>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_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="E1:17_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_1_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/><a NAME="E3:17_1_1_1_1_1_2_1_1_1"/><b>then </b>
 <a href="supinf_2.html#K1">0.</a>  <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>2</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green">Lemma10</font></i></a></i>;<br/><a NAME="E4:17_1_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_2_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_2_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green">Lemma10</font></i></a></i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_2_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_1_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_2_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_2_1_1_3"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_2_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_1_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T134">XREAL_1:134</a></i>;<br/><a NAME="E3:17_1_1_1_1_1_2_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E18"><i><font color="Green">Lemma11</font></i></a></i>;<br/><a NAME="E4:17_1_1_1_1_1_2_1_1_3"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_2_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_2_1_1_3"><i><font color="Green">E22</font></i></a>, <a class="txt" href="extreal1.html#E18"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_2_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_2_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:17_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_3"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">c<sub>3</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="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>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1_1_1_3_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_3_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_1_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T134">XREAL_1:134</a></i>;<br/><a NAME="E3:17_1_1_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E18"><i><font color="Green">Lemma11</font></i></a></i>;<br/><a NAME="E4:17_1_1_1_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_1_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_3_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_3_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green">Lemma10</font></i></a></i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_3_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_3_1_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_1_1_1_3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_3_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_3_1_1_3"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_1_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/><a NAME="E3:17_1_1_1_1_1_3_1_1_3"/><b>then </b>
 <a href="supinf_2.html#K1">0.</a>  <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>2</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green">Lemma10</font></i></a></i>;<br/><a NAME="E4:17_1_1_1_1_1_3_1_1_3"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_1_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_3_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_3_1_1_3"><i><font color="Green">E22</font></i></a>, <a class="txt" href="extreal1.html#E18"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_3_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_3_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:17_1_1_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_4"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</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; <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>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_4"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1_1_1_4_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_4_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_1_1_1_4"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T134">XREAL_1:134</a></i>;<br/><a NAME="E3:17_1_1_1_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E18"><i><font color="Green">Lemma11</font></i></a></i>;<br/><a NAME="E4:17_1_1_1_1_1_4_1_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_1_1_1_4"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_4_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_4_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green">Lemma10</font></i></a></i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_4_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_4"><i><font color="Green">E20</font></i></a>, <a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_4_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_4_1_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_1_1_1_4_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_4_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_4_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_4_1_1_3"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_4_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_1_1_1_4"><i><font color="Green">E21</font></i></a>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/><a NAME="E3:17_1_1_1_1_1_4_1_1_3"/><b>then </b>
 <a href="supinf_2.html#K1">0.</a>  <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>2</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green">Lemma10</font></i></a></i>;<br/><a NAME="E4:17_1_1_1_1_1_4_1_1_3"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_1_1_1_4"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_4_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_4_1_1_3"><i><font color="Green">E22</font></i></a>, <a class="txt" href="extreal1.html#E18"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_4_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_4"><i><font color="Green">E20</font></i></a>, <a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_4_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:17_1_1_1_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_5"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_5"/><i><font color="Green">E20</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</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="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>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_5"/><b>then </b><i><font color="Green">E21</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1_1_1_5_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_5_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_5_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_1_1_1_5"><i><font color="Green">E21</font></i></a>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/><a NAME="E3:17_1_1_1_1_1_5_1_1_1"/><b>then </b>
 <a href="supinf_2.html#K1">0.</a>  <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>2</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green">Lemma10</font></i></a></i>;<br/><a NAME="E4:17_1_1_1_1_1_5_1_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_1_1_1_5"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_5_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_5_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green">Lemma10</font></i></a></i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_5_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_5"><i><font color="Green">E20</font></i></a>, <a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_5_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_5_1_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_5_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_1_1_1_5_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_5_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_5_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_5_1_1_3"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_5_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_1_1_1_5"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xreal_1.html#T134">XREAL_1:134</a></i>;<br/><a NAME="E3:17_1_1_1_1_1_5_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E18"><i><font color="Green">Lemma11</font></i></a></i>;<br/><a NAME="E4:17_1_1_1_1_1_5_1_1_3"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_1_1_1_5"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_5_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_5_1_1_3"><i><font color="Green">E22</font></i></a>, <a class="txt" href="extreal1.html#E18"><i><font color="Green">Lemma11</font></i></a></i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_5_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_5"><i><font color="Green">E20</font></i></a>, <a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_5_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:17_1_1_1_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_6"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_6"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &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>  <a href="supinf_2.html#K1">0.</a>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_1_1_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_6"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_7"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_7"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &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>  <a href="supinf_2.html#K1">0.</a>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_1_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_7"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:17_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_2"/><i><font color="Green">E18</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <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"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_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="E1:17_1_1_2_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <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>  )
 </span><b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_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#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_2"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_2"/><i><font color="Green">E19</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_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="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_2_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_2_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_3"/>
<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> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_2_1_1_3"/><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#E1:17_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_2_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_3"/><i><font color="Green">E18</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <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"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_3_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <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>  )
 </span><b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_3_1_1_1"/><i><font color="Green">E19</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_3"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_1_3_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</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#E1:17_1_1_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_3_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_3_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_3_1_1_2"/><i><font color="Green">E19</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_1_3_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>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_3_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_3_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_3_1_1_3"/><i><font color="Green">E19</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> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_3_1_1_3"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_3"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_1_3_1_1_3"/>
<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>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_3_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_3_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_3"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_3_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_4"><b>suppose </b></a><a NAME="E1:17_1_1_4"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</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; <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <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"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_4_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <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>  )
 </span><b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_4_1_1_1"/><i><font color="Green">E19</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_4_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_4"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_4_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_4_1_1_2"/><i><font color="Green">E19</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_4_1_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_1_4_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>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_4_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_4_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_4_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_4_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_4_1_1_3"/><i><font color="Green">E19</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> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_4_1_1_3"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_1_4_1_1_3"/>
<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#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_4"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_4_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_4_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_4"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_4_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_5"><b>suppose </b></a><a NAME="E1:17_1_1_5"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <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"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_5_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <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>  )
 </span><b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_5_1_1_1"/><i><font color="Green">E19</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_5_1_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_5"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_1_5_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>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_5"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_5_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_5_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_5"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_5_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_5_1_1_2"/><i><font color="Green">E19</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_5_1_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_1_5_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>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_5_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_5_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_5_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_5_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_5_1_1_3"/><i><font color="Green">E19</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> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_5_1_1_3"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_5"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_1_5_1_1_3"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</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#E1:17_1_1_5"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_5_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_5_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_5"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_5_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_6"><b>suppose </b></a><a NAME="E1:17_1_1_6"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</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; <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <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"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_6_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <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>  )
 </span><b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_6_1_1_1"/><i><font color="Green">E19</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_6_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#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_6"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_6_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_6"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_6_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_6_1_1_2"/><i><font color="Green">E19</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_6_1_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_1_6_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>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_6_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_6_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_6_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_6_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_6_1_1_3"/><i><font color="Green">E19</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> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_6_1_1_3"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_6"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_1_6_1_1_3"/>
<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#E1:17_1_1_6"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_6_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_6_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_6"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_6_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_7"><b>suppose </b></a><a NAME="E1:17_1_1_7"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <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"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_7_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <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>  )
 </span><b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_7_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_7_1_1_1"/><i><font color="Green">E19</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_7_1_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_7"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_1_7_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>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_7"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_7_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_7_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_7"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_7_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_7_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_7_1_1_2"/><i><font color="Green">E19</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_7_1_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_1_7_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>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_7_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_7_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_7_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_7_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_7_1_1_3"/><i><font color="Green">E19</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> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_7_1_1_3"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_7"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_1_7_1_1_3"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</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#E1:17_1_1_7"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_7_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_7_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_7"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_7_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_8"><b>suppose </b></a><a NAME="E1:17_1_1_8"/><i><font color="Green">E18</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <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"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_8_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <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>  )
 </span><b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_8_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_8_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_8_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#E1:17_1_1_8"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_8_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_8"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_8_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_8_1_1_2"/><i><font color="Green">E19</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_8_1_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_1_8_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>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_8_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_8_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_8_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_8_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_8_1_1_3"/>
<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> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_8_1_1_3"/><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#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_8"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_8_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_8"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_9"><b>suppose </b></a><a NAME="E1:17_1_1_9"/><i><font color="Green">E18</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <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"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_9_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_9_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_9_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <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>  )
 </span><b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_9_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_9_1_1_1"/><i><font color="Green">E19</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_9_1_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_9"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_1_9_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</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#E1:17_1_1_9"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_9_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_9_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_9"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_9_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_9_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_9_1_1_2"/><i><font color="Green">E19</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_9_1_1_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_1_9_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>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_9_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_9_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_9_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_9_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_9_1_1_3"/><i><font color="Green">E19</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> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_9_1_1_3"/><b>then </b><i><font color="Green">E20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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#E1:17_1_1_9"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:17_1_1_9_1_1_3"/>
<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>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_9"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_9_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="ref" href="extreal1.html#T21" target="_self">Th21</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_9_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_9"><i><font color="Green">E18</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_9_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_10"><b>suppose </b></a><a NAME="E1:17_1_1_10"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_10"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_10"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_10"><i><font color="Green">E18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_11"><b>suppose </b></a><a NAME="E1:17_1_1_11"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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_2.html#K1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_11"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_11"><i><font color="Green">E18</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="E2:17"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>


</div>
