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



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_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>  ) 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>  ) 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>  ) 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>  ) 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>  ) 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>  ) 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>  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="19_1_1_1"><b>suppose </b></a><a NAME="E1:19_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>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:19_1_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</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>3</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E4:19_1_1_1"/><i><font color="Green">E21</font></i>: 
(  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="supinf_2.html#K3">-</a> <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="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="extreal1.html#E3:19_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="supinf_2.html#T3">SUPINF_2:3</a></i>;<br/><a NAME="E5:19_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="extreal1.html#E3:19_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="E6:19_1_1_1"/>
(  <a href="supinf_2.html#K3">-</a> <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="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"><a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="supinf_2.html#K3">-</a> <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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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#E4:19_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="19_1_1_2"><b>suppose </b></a><a NAME="E1:19_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>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>  ) 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>  ) 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>  ) 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>  )
 ;<br/><div class="add"><a NAME="E2:19_1_1_2"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="supinf_2.html#K3">-</a> <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="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E3:19_1_1_2"/><i><font color="Green">E22</font></i>: 
( (  <a href="supinf_2.html#K3">-</a> <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>  ) 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;  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</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>  <a href="supinf_2.html#K3">-</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>  ) 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;  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</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#E1:19_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#T4" target="_self">Th4</a>, <a class="ref" href="extreal1.html#T24" target="_self">Th24</a>, <a class="ref" href="supinf_2.html#T17">SUPINF_2:17</a></i>;<br/><a NAME="E4:19_1_1_2"/>
( (  <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;  <a href="supinf_2.html#K3">-</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#K3">-</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>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="supinf_2.html#K3">-</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>  <a href="supinf_2.html#K3">-</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>  ) )
 <b>by </b><i><a class="txt" href="extreal1.html#E1:19_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#T4" target="_self">Th4</a>, <a class="ref" href="extreal1.html#T24" target="_self">Th24</a>, <a class="ref" href="supinf_2.html#T17">SUPINF_2:17</a></i>;<br/><b>hence </b><a NAME="E5:19_1_1_2"/>
(  <a href="supinf_2.html#K3">-</a> <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="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"><a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="supinf_2.html#K3">-</a> <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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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#E2:19_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="extreal1.html#E3:19_1_1_2"><i><font color="Green">E22</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="19_1_1_3"><b>suppose </b></a><a NAME="E1:19_1_1_3"/><i><font color="Green">E20</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>  ) 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>  ) 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>  ) 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>  )
 ;<br/><div class="add"><a NAME="E2:19_1_1_3"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="supinf_2.html#K3">-</a> <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="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E3:19_1_1_3"/><i><font color="Green">E22</font></i>: 
( (  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K3">-</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>  ) 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;  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) or (  <a href="supinf_2.html#K3">-</a> <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>  ) 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;  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</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#E1:19_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#T4" target="_self">Th4</a>, <a class="ref" href="extreal1.html#T24" target="_self">Th24</a>, <a class="ref" href="supinf_2.html#T17">SUPINF_2:17</a></i>;<br/><a NAME="E4:19_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>  &amp;  <a href="supinf_2.html#K3">-</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>  <a href="supinf_2.html#K3">-</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>  ) 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;  <a href="supinf_2.html#K3">-</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#K3">-</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>  &amp; <font color="Maroon">c<sub>1</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#E1:19_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="extreal1.html#T4" target="_self">Th4</a>, <a class="ref" href="extreal1.html#T24" target="_self">Th24</a>, <a class="ref" href="supinf_2.html#T17">SUPINF_2:17</a></i>;<br/><b>hence </b><a NAME="E5:19_1_1_3"/>
(  <a href="supinf_2.html#K3">-</a> <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="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"><a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="supinf_2.html#K3">-</a> <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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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#E2:19_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="extreal1.html#E3:19_1_1_3"><i><font color="Green">E22</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="19_1_1_4"><b>suppose </b></a><a NAME="E1:19_1_1_4"/>
( ( <font color="Maroon">c<sub>1</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>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_1_1_4"/>
(  <a href="supinf_2.html#K3">-</a> <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="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"><a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="supinf_2.html#K3">-</a> <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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self">Def1</a>, <a class="ref" href="extreal1.html#T24" target="_self">Th24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:19"/>
(  <a href="supinf_2.html#K3">-</a> <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="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"><a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="supinf_2.html#K3">-</a> <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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>


</div>
