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

<b>assume </b><a NAME="E1:23"/><i><font color="Green">E21</font></i>: 
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#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="xxreal_0.html#NR3" target="_self">&lt;</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>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</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#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#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="xxreal_0.html#NR3" target="_self">&lt;</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>  or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</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#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 </span><b>by </b><i><a class="txt" href="extreal1.html#E1:23"><i><font color="Green">E21</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1"><b>case </b></a><a NAME="E1:23_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:23_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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E34"><i><font color="Green">Lemma20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2"><b>case </b></a><a NAME="E1:23_1_1_2"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</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#T9" target="_self">Th9</a>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a></i>;<br/><a NAME="E3:23_1_1_2"/><b>then </b><i><font color="Green">E23</font></i>: 
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_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="E1:23_1_1_2_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 </span><b>by </b><i><a class="ref" href="supinf_2.html#T2">SUPINF_2:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:23_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:23_1_1_2_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="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>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#E1:23_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#T19">SUPINF_2:19</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_2_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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#T9" target="_self">Th9</a>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:23_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:23_1_1_2_1_1_2"/><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="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>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#E1:23_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#T19">SUPINF_2:19</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_2_1_1_2"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#T9" target="_self">Th9</a>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_1_3"><b>case </b></a><a NAME="E1:23_1_1_2_1_1_3"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_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="E1:23_1_1_2_1_1_3_1_1"/>
( (  <a href="supinf_1.html#K2">-infty</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>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</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 (  <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>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 </span><b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_2_1_1_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="extreal1.html#T16" target="_self">Th16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_1_3_1_1_1"><b>case </b></a><a NAME="E1:23_1_1_2_1_1_3_1_1_1"/>
(  <a href="supinf_1.html#K2">-infty</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>1</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="E2:23_1_1_2_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="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>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#E1:23_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_2_1_1_3_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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#T9" target="_self">Th9</a>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_1_3_1_1_2"><b>case </b></a><a NAME="E1:23_1_1_2_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:23_1_1_2_1_1_3_1_1_2"/><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="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>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="E3:23_1_1_2_1_1_3_1_1_2"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2_1_1_3_1_1_3"><b>case </b></a><a NAME="E1:23_1_1_2_1_1_3_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>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_2_1_1_3_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="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>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#E1:23_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_2_1_1_3_1_1_3"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#T9" target="_self">Th9</a>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:23_1_1_2_1_1_3"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:23_1_1_2"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3"><b>case </b></a><a NAME="E1:23_1_1_3"/><i><font color="Green">E22</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="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_3"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</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="ref" href="extreal1.html#T15" target="_self">Th15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_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="E1:23_1_1_3_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 </span><b>by </b><i><a class="ref" href="supinf_2.html#T2">SUPINF_2:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_1"><b>case </b></a><a NAME="E1:23_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:23_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="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>3</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3"><i><font color="Green">E22</font></i></a>, <a class="txt" href="extreal1.html#E2:23_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_3_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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_2"><b>case </b></a><a NAME="E1:23_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:23_1_1_3_1_1_2"/><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="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>3</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3"><i><font color="Green">E22</font></i></a>, <a class="txt" href="extreal1.html#E2:23_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_3_1_1_2"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_3"><b>case </b></a><a NAME="E1:23_1_1_3_1_1_3"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_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="E1:23_1_1_3_1_1_3_1_1"/>
( (  <a href="supinf_1.html#K2">-infty</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>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</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 (  <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>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 </span><b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="extreal1.html#T16" target="_self">Th16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_3_1_1_1"><b>case </b></a><a NAME="E1:23_1_1_3_1_1_3_1_1_1"/><i><font color="Green">E25</font></i>: 
(  <a href="supinf_1.html#K2">-infty</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>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_3_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="E1:23_1_1_3_1_1_3_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <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_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <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#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) )
 </span><b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3"><i><font color="Green">E22</font></i></a>, <a class="ref" href="supinf_2.html#T2">SUPINF_2:2</a>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_3_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:23_1_1_3_1_1_3_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <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="E2:23_1_1_3_1_1_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
( <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>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</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#E1:23_1_1_3_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E3:23_1_1_3_1_1_3_1_1_1_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="supinf_2.html#K2">+</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>3</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="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_3_1_1_3_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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="extreal1.html#E2:23_1_1_3_1_1_3_1_1_1_1_1_1"><i><font color="Green">E26</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="23_1_1_3_1_1_3_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:23_1_1_3_1_1_3_1_1_1_1_1_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_3_1_1_3_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</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="ref" href="supinf_1.html#T1">SUPINF_1:1</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E3:23_1_1_3_1_1_3_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E4:23_1_1_3_1_1_3_1_1_1_1_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_1_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:23_1_1_3_1_1_3_1_1_1_1_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3"><i><font color="Green">E24</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:23_1_1_3_1_1_3_1_1_1_1_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_1_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E9:23_1_1_3_1_1_3_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E6:23_1_1_3_1_1_3_1_1_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="extreal1.html#E8:23_1_1_3_1_1_3_1_1_1_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E10:23_1_1_3_1_1_3_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><b>hence </b><a NAME="E11:23_1_1_3_1_1_3_1_1_1_1_1_2"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_3_1_1_3_1_1_1_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="extreal1.html#E4:23_1_1_3_1_1_3_1_1_1_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_3_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:23_1_1_3_1_1_3_1_1_1_1_1_3"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <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="E2:23_1_1_3_1_1_3_1_1_1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</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="ref" href="supinf_1.html#T1">SUPINF_1:1</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E3:23_1_1_3_1_1_3_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E27</font></i>: 
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E4:23_1_1_3_1_1_3_1_1_1_1_1_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_1_1_1_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:23_1_1_3_1_1_3_1_1_1_1_1_3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3"><i><font color="Green">E24</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:23_1_1_3_1_1_3_1_1_1_1_1_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_1_1_1_3"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E9:23_1_1_3_1_1_3_1_1_1_1_1_3"/>
<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>
 <b>by </b><i><a class="txt" href="extreal1.html#E6:23_1_1_3_1_1_3_1_1_1_1_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="extreal1.html#E8:23_1_1_3_1_1_3_1_1_1_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E10:23_1_1_3_1_1_3_1_1_1_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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><b>hence </b><a NAME="E11:23_1_1_3_1_1_3_1_1_1_1_1_3"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_3_1_1_3_1_1_1_1_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="extreal1.html#E4:23_1_1_3_1_1_3_1_1_1_1_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_3_1_1_1_1_1_4"><b>case </b></a><a NAME="E1:23_1_1_3_1_1_3_1_1_1_1_1_4"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:23_1_1_3_1_1_3_1_1_1_1_1_4"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:23_1_1_3_1_1_3_1_1_1_1_1_4"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_1_1_1_4"><i><font color="Green">E26</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:23_1_1_3_1_1_3_1_1_1_1_1_4"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E8:23_1_1_3_1_1_3_1_1_1_1_1_4"/>
( <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>6</sub></font> <a href="real_1.html#K4">*</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>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>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_3_1_1_3_1_1_1_1_1_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="extreal1.html#E5:23_1_1_3_1_1_3_1_1_1_1_1_4"><i><font color="Green">E28</font></i></a>, <a class="txt" href="extreal1.html#E7:23_1_1_3_1_1_3_1_1_1_1_1_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/><a NAME="E9:23_1_1_3_1_1_3_1_1_1_1_1_4"/><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="supinf_2.html#K2">+</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>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="extreal1.html#E7:23_1_1_3_1_1_3_1_1_1_1_1_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/><b>hence </b><a NAME="E10:23_1_1_3_1_1_3_1_1_1_1_1_4"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:23_1_1_3_1_1_3_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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_3_1_1_2"><b>case </b></a><a NAME="E1:23_1_1_3_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:23_1_1_3_1_1_3_1_1_2"/><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="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>3</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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="E3:23_1_1_3_1_1_3_1_1_2"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_3_1_1_3"><b>case </b></a><a NAME="E1:23_1_1_3_1_1_3_1_1_3"/><i><font color="Green">E25</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>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3_1_1_3_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_3_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="E1:23_1_1_3_1_1_3_1_1_3_1_1"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <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#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) )
 </span><b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3"><i><font color="Green">E22</font></i></a>, <a class="ref" href="supinf_2.html#T2">SUPINF_2:2</a>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_3_1_1_3_1_1_1"><b>case </b></a><a NAME="E1:23_1_1_3_1_1_3_1_1_3_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <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="E2:23_1_1_3_1_1_3_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
( <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>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</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#E1:23_1_1_3_1_1_3_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E3:23_1_1_3_1_1_3_1_1_3_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="supinf_2.html#K2">+</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>3</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="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_3_1_1_3_1_1_3_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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="extreal1.html#E2:23_1_1_3_1_1_3_1_1_3_1_1_1"><i><font color="Green">E26</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="23_1_1_3_1_1_3_1_1_3_1_1_2"><b>case </b></a><a NAME="E1:23_1_1_3_1_1_3_1_1_3_1_1_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_3_1_1_3_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</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="ref" href="supinf_1.html#T1">SUPINF_1:1</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E3:23_1_1_3_1_1_3_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E4:23_1_1_3_1_1_3_1_1_3_1_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</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#E1:23_1_1_3_1_1_3_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_3_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:23_1_1_3_1_1_3_1_1_3_1_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3"><i><font color="Green">E24</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:23_1_1_3_1_1_3_1_1_3_1_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_3_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E9:23_1_1_3_1_1_3_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E6:23_1_1_3_1_1_3_1_1_3_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="extreal1.html#E8:23_1_1_3_1_1_3_1_1_3_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E10:23_1_1_3_1_1_3_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><b>hence </b><a NAME="E11:23_1_1_3_1_1_3_1_1_3_1_1_2"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_3_1_1_3_1_1_3_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="extreal1.html#E4:23_1_1_3_1_1_3_1_1_3_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_3_1_1_3_1_1_3"><b>case </b></a><a NAME="E1:23_1_1_3_1_1_3_1_1_3_1_1_3"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <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="E2:23_1_1_3_1_1_3_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</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="ref" href="supinf_1.html#T1">SUPINF_1:1</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E3:23_1_1_3_1_1_3_1_1_3_1_1_3"/><b>then </b><i><font color="Green">E27</font></i>: 
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E4:23_1_1_3_1_1_3_1_1_3_1_1_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_3_1_1_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:23_1_1_3_1_1_3_1_1_3_1_1_3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3"><i><font color="Green">E24</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:23_1_1_3_1_1_3_1_1_3_1_1_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_3_1_1_3"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E9:23_1_1_3_1_1_3_1_1_3_1_1_3"/>
<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>
 <b>by </b><i><a class="txt" href="extreal1.html#E6:23_1_1_3_1_1_3_1_1_3_1_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="extreal1.html#E8:23_1_1_3_1_1_3_1_1_3_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E10:23_1_1_3_1_1_3_1_1_3_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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><b>hence </b><a NAME="E11:23_1_1_3_1_1_3_1_1_3_1_1_3"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_3_1_1_3_1_1_3_1_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="extreal1.html#E4:23_1_1_3_1_1_3_1_1_3_1_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_3_1_1_3_1_1_3_1_1_4"><b>case </b></a><a NAME="E1:23_1_1_3_1_1_3_1_1_3_1_1_4"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:23_1_1_3_1_1_3_1_1_3_1_1_4"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:23_1_1_3_1_1_3_1_1_3_1_1_4"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3_1_1_3_1_1_4"><i><font color="Green">E26</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:23_1_1_3_1_1_3_1_1_3_1_1_4"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_3_1_1_3"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E8:23_1_1_3_1_1_3_1_1_3_1_1_4"/>
( <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>6</sub></font> <a href="real_1.html#K4">*</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>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>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_3_1_1_3_1_1_3_1_1_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="extreal1.html#E5:23_1_1_3_1_1_3_1_1_3_1_1_4"><i><font color="Green">E28</font></i></a>, <a class="txt" href="extreal1.html#E7:23_1_1_3_1_1_3_1_1_3_1_1_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/><a NAME="E9:23_1_1_3_1_1_3_1_1_3_1_1_4"/><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="supinf_2.html#K2">+</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>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="extreal1.html#E7:23_1_1_3_1_1_3_1_1_3_1_1_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/><b>hence </b><a NAME="E10:23_1_1_3_1_1_3_1_1_3_1_1_4"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:23_1_1_3_1_1_3_1_1_3"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:23_1_1_3_1_1_3"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:23_1_1_3"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_4"><b>case </b></a><a NAME="E1:23_1_1_4"/>
<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="E2:23_1_1_4"/><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="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self">Def1</a>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_4"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_5"><b>case </b></a><a NAME="E1:23_1_1_5"/>
<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="E2:23_1_1_5"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self">Def1</a>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_5"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_6"><b>case </b></a><a NAME="E1:23_1_1_6"/><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> <font color="Maroon">c<sub>2</sub></font> &amp;  <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="E2:23_1_1_6"/><b>then </b><i><font color="Green">E23</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> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="extreal1.html#T14" target="_self">Th14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_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="E1:23_1_1_6_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 </span><b>by </b><i><a class="ref" href="supinf_2.html#T2">SUPINF_2:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_6_1_1_1"><b>case </b></a><a NAME="E1:23_1_1_6_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:23_1_1_6_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="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>3</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6"><i><font color="Green">E22</font></i></a>, <a class="txt" href="extreal1.html#E2:23_1_1_6"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_6_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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_6_1_1_2"><b>case </b></a><a NAME="E1:23_1_1_6_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:23_1_1_6_1_1_2"/><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="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>3</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6"><i><font color="Green">E22</font></i></a>, <a class="txt" href="extreal1.html#E2:23_1_1_6"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_6_1_1_2"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_6_1_1_3"><b>case </b></a><a NAME="E1:23_1_1_6_1_1_3"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_6_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_6_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="E1:23_1_1_6_1_1_3_1_1"/>
( (  <a href="supinf_1.html#K2">-infty</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>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</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 (  <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>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 </span><b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="extreal1.html#T16" target="_self">Th16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_6_1_1_3_1_1_1"><b>case </b></a><a NAME="E1:23_1_1_6_1_1_3_1_1_1"/><i><font color="Green">E25</font></i>: 
(  <a href="supinf_1.html#K2">-infty</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>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_6_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_6_1_1_3_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="E1:23_1_1_6_1_1_3_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <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="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <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="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) )
 </span><b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6"><i><font color="Green">E22</font></i></a>, <a class="ref" href="supinf_2.html#T2">SUPINF_2:2</a>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_6_1_1_3_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:23_1_1_6_1_1_3_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <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="E2:23_1_1_6_1_1_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
( <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>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</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#E1:23_1_1_6_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E3:23_1_1_6_1_1_3_1_1_1_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="supinf_2.html#K2">+</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>3</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="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_6_1_1_3_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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="extreal1.html#E2:23_1_1_6_1_1_3_1_1_1_1_1_1"><i><font color="Green">E26</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="23_1_1_6_1_1_3_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:23_1_1_6_1_1_3_1_1_1_1_1_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_6_1_1_3_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</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="ref" href="supinf_1.html#T6">SUPINF_1:6</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E3:23_1_1_6_1_1_3_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E4:23_1_1_6_1_1_3_1_1_1_1_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</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#E1:23_1_1_6_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_1_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:23_1_1_6_1_1_3_1_1_1_1_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3"><i><font color="Green">E24</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:23_1_1_6_1_1_3_1_1_1_1_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_1_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E9:23_1_1_6_1_1_3_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E6:23_1_1_6_1_1_3_1_1_1_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="extreal1.html#E8:23_1_1_6_1_1_3_1_1_1_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E10:23_1_1_6_1_1_3_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><b>hence </b><a NAME="E11:23_1_1_6_1_1_3_1_1_1_1_1_2"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_6_1_1_3_1_1_1_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="extreal1.html#E4:23_1_1_6_1_1_3_1_1_1_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_6_1_1_3_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:23_1_1_6_1_1_3_1_1_1_1_1_3"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <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="E2:23_1_1_6_1_1_3_1_1_1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</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="ref" href="supinf_1.html#T6">SUPINF_1:6</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E3:23_1_1_6_1_1_3_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E27</font></i>: 
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E4:23_1_1_6_1_1_3_1_1_1_1_1_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_1_1_1_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:23_1_1_6_1_1_3_1_1_1_1_1_3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3"><i><font color="Green">E24</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:23_1_1_6_1_1_3_1_1_1_1_1_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_1_1_1_3"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E9:23_1_1_6_1_1_3_1_1_1_1_1_3"/>
<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>
 <b>by </b><i><a class="txt" href="extreal1.html#E6:23_1_1_6_1_1_3_1_1_1_1_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="extreal1.html#E8:23_1_1_6_1_1_3_1_1_1_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E10:23_1_1_6_1_1_3_1_1_1_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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><b>hence </b><a NAME="E11:23_1_1_6_1_1_3_1_1_1_1_1_3"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_6_1_1_3_1_1_1_1_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="extreal1.html#E4:23_1_1_6_1_1_3_1_1_1_1_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_6_1_1_3_1_1_1_1_1_4"><b>case </b></a><a NAME="E1:23_1_1_6_1_1_3_1_1_1_1_1_4"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:23_1_1_6_1_1_3_1_1_1_1_1_4"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:23_1_1_6_1_1_3_1_1_1_1_1_4"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_1_1_1_4"><i><font color="Green">E26</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:23_1_1_6_1_1_3_1_1_1_1_1_4"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E8:23_1_1_6_1_1_3_1_1_1_1_1_4"/>
( <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>6</sub></font> <a href="real_1.html#K4">*</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>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>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_6_1_1_3_1_1_1_1_1_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="extreal1.html#E5:23_1_1_6_1_1_3_1_1_1_1_1_4"><i><font color="Green">E28</font></i></a>, <a class="txt" href="extreal1.html#E7:23_1_1_6_1_1_3_1_1_1_1_1_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/><a NAME="E9:23_1_1_6_1_1_3_1_1_1_1_1_4"/><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="supinf_2.html#K2">+</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>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="extreal1.html#E7:23_1_1_6_1_1_3_1_1_1_1_1_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/><b>hence </b><a NAME="E10:23_1_1_6_1_1_3_1_1_1_1_1_4"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:23_1_1_6_1_1_3_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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_6_1_1_3_1_1_2"><b>case </b></a><a NAME="E1:23_1_1_6_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:23_1_1_6_1_1_3_1_1_2"/><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="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>3</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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="E3:23_1_1_6_1_1_3_1_1_2"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_6_1_1_3_1_1_3"><b>case </b></a><a NAME="E1:23_1_1_6_1_1_3_1_1_3"/><i><font color="Green">E25</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>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_6_1_1_3_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_6_1_1_3_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="E1:23_1_1_6_1_1_3_1_1_3_1_1"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) or ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <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="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) )
 </span><b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6"><i><font color="Green">E22</font></i></a>, <a class="ref" href="supinf_2.html#T2">SUPINF_2:2</a>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_6_1_1_3_1_1_3_1_1_1"><b>case </b></a><a NAME="E1:23_1_1_6_1_1_3_1_1_3_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <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="E2:23_1_1_6_1_1_3_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
( <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>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</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#E1:23_1_1_6_1_1_3_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E3:23_1_1_6_1_1_3_1_1_3_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="supinf_2.html#K2">+</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>3</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="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_6_1_1_3_1_1_3_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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="extreal1.html#E2:23_1_1_6_1_1_3_1_1_3_1_1_1"><i><font color="Green">E26</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="23_1_1_6_1_1_3_1_1_3_1_1_2"><b>case </b></a><a NAME="E1:23_1_1_6_1_1_3_1_1_3_1_1_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_6_1_1_3_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</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="ref" href="supinf_1.html#T6">SUPINF_1:6</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E3:23_1_1_6_1_1_3_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E4:23_1_1_6_1_1_3_1_1_3_1_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_3_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:23_1_1_6_1_1_3_1_1_3_1_1_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3"><i><font color="Green">E24</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:23_1_1_6_1_1_3_1_1_3_1_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_3_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E9:23_1_1_6_1_1_3_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E6:23_1_1_6_1_1_3_1_1_3_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="extreal1.html#E8:23_1_1_6_1_1_3_1_1_3_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E10:23_1_1_6_1_1_3_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><b>hence </b><a NAME="E11:23_1_1_6_1_1_3_1_1_3_1_1_2"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_6_1_1_3_1_1_3_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="extreal1.html#E4:23_1_1_6_1_1_3_1_1_3_1_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_6_1_1_3_1_1_3_1_1_3"><b>case </b></a><a NAME="E1:23_1_1_6_1_1_3_1_1_3_1_1_3"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <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="E2:23_1_1_6_1_1_3_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</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="ref" href="supinf_1.html#T6">SUPINF_1:6</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><a NAME="E3:23_1_1_6_1_1_3_1_1_3_1_1_3"/><b>then </b><i><font color="Green">E27</font></i>: 
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E4:23_1_1_6_1_1_3_1_1_3_1_1_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_3_1_1_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:23_1_1_6_1_1_3_1_1_3_1_1_3"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3"><i><font color="Green">E24</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:23_1_1_6_1_1_3_1_1_3_1_1_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_3_1_1_3"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E9:23_1_1_6_1_1_3_1_1_3_1_1_3"/>
<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>
 <b>by </b><i><a class="txt" href="extreal1.html#E6:23_1_1_6_1_1_3_1_1_3_1_1_3"><i><font color="Green">E29</font></i></a>, <a class="txt" href="extreal1.html#E8:23_1_1_6_1_1_3_1_1_3_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E10:23_1_1_6_1_1_3_1_1_3_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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><b>hence </b><a NAME="E11:23_1_1_6_1_1_3_1_1_3_1_1_3"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_6_1_1_3_1_1_3_1_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="extreal1.html#E4:23_1_1_6_1_1_3_1_1_3_1_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_6_1_1_3_1_1_3_1_1_4"><b>case </b></a><a NAME="E1:23_1_1_6_1_1_3_1_1_3_1_1_4"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:23_1_1_6_1_1_3_1_1_3_1_1_4"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:23_1_1_6_1_1_3_1_1_3_1_1_4"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3_1_1_3_1_1_4"><i><font color="Green">E26</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:23_1_1_6_1_1_3_1_1_3_1_1_4"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_6_1_1_3"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E8:23_1_1_6_1_1_3_1_1_3_1_1_4"/>
( <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>6</sub></font> <a href="real_1.html#K4">*</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>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>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_6_1_1_3_1_1_3_1_1_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="extreal1.html#E5:23_1_1_6_1_1_3_1_1_3_1_1_4"><i><font color="Green">E28</font></i></a>, <a class="txt" href="extreal1.html#E7:23_1_1_6_1_1_3_1_1_3_1_1_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/><a NAME="E9:23_1_1_6_1_1_3_1_1_3_1_1_4"/><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="supinf_2.html#K2">+</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>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="extreal1.html#E7:23_1_1_6_1_1_3_1_1_3_1_1_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self">Th13</a>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/><b>hence </b><a NAME="E10:23_1_1_6_1_1_3_1_1_3_1_1_4"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:23_1_1_6_1_1_3_1_1_3"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:23_1_1_6_1_1_3"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:23_1_1_6"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7"><b>case </b></a><a NAME="E1:23_1_1_7"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <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="E2:23_1_1_7"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K2">+</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#T9" target="_self">Th9</a>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a></i>;<br/><a NAME="E3:23_1_1_7"/><b>then </b><i><font color="Green">E23</font></i>: 
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_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="E1:23_1_1_7_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 </span><b>by </b><i><a class="ref" href="supinf_2.html#T2">SUPINF_2:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_1_1_1"><b>case </b></a><a NAME="E1:23_1_1_7_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:23_1_1_7_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="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>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#E1:23_1_1_7"><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#T19">SUPINF_2:19</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_7_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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_7"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#T9" target="_self">Th9</a>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_1_1_2"><b>case </b></a><a NAME="E1:23_1_1_7_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:23_1_1_7_1_1_2"/><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="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>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#E1:23_1_1_7"><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#T19">SUPINF_2:19</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_7_1_1_2"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_7"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#T9" target="_self">Th9</a>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_1_1_3"><b>case </b></a><a NAME="E1:23_1_1_7_1_1_3"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_7_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_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="E1:23_1_1_7_1_1_3_1_1"/>
( (  <a href="supinf_1.html#K2">-infty</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>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</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 (  <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>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 </span><b>by </b><i><a class="txt" href="extreal1.html#E1:23_1_1_7_1_1_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="extreal1.html#T16" target="_self">Th16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_1_1_3_1_1_1"><b>case </b></a><a NAME="E1:23_1_1_7_1_1_3_1_1_1"/>
(  <a href="supinf_1.html#K2">-infty</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>1</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="E2:23_1_1_7_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="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>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#E1:23_1_1_7"><i><font color="Green">E22</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_7_1_1_3_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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_7"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#T9" target="_self">Th9</a>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_1_1_3_1_1_2"><b>case </b></a><a NAME="E1:23_1_1_7_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:23_1_1_7_1_1_3_1_1_2"/><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="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>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="E3:23_1_1_7_1_1_3_1_1_2"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_7"><i><font color="Green">E23</font></i></a>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_7_1_1_3_1_1_3"><b>case </b></a><a NAME="E1:23_1_1_7_1_1_3_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>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><a NAME="E2:23_1_1_7_1_1_3_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="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>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#E1:23_1_1_7"><i><font color="Green">E22</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1_7_1_1_3_1_1_3"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:23_1_1_7"><i><font color="Green">E23</font></i></a>, <a class="ref" href="extreal1.html#T9" target="_self">Th9</a>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:23_1_1_7_1_1_3"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:23_1_1_7"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:23"/>
<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="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</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="supinf_2.html#K2">+</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>3</sub></font></span>)</span>
 ;<br/>


</div>
