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

<b>let </b><font color="Maroon">x</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1"/>
(  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="real_1.html#K4">*</a> <font color="Olive">b</font> ) or (  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or (  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or ( <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or ( <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or ( <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) or ( <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) or (  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) or (  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) or ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) or ( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="real_1.html#K4">*</a> <font color="Olive">b</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:17_1_1_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> )
 ;<br/><a NAME="E4:17_1_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1_1_1"/>
(  ex <font color="Olive">b1</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">b1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">b1</font> <a href="real_1.html#K4">*</a> <font color="Olive">c</font> ) or (  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or ( <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or ( <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) or (  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) or ( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) or ( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) )
 </span><b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green">E19</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_1"/>
 ex <font color="Olive">b1</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">b1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Olive">b1</font> <a href="real_1.html#K4">*</a> <font color="Olive">c</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">b1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:17_1_1_1_1_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font> )
 ;<br/><a NAME="E4:17_1_1_1_1_1_1"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, <a class="txt" href="extreal1.html#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, </i>;<br/><a NAME="E5:17_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, <a class="txt" href="extreal1.html#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_2"/><i><font color="Green">E30</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/><a NAME="E3:17_1_1_1_1_1_2_1_1_1"/><b>then </b>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E4:17_1_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</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#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, </i>;<br/><a NAME="E5:17_1_1_1_1_1_2_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, <a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</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#T7" target="_self">Th7</a>, , <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_1_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, <a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, , <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_2_1_1_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_2_1_1_3"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="xreal_1.html#T134">XREAL_1:134</a></i>;<br/><a NAME="E3:17_1_1_1_1_1_2_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</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#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E4:17_1_1_1_1_1_2_1_1_3"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</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#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, </i>;<br/><a NAME="E5:17_1_1_1_1_1_2_1_1_3"/>
<font color="Maroon">x</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#T7" target="_self">Th7</a>, <a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_2_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:17_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_3"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1_1_1_3_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_3_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="xreal_1.html#T134">XREAL_1:134</a></i>;<br/><a NAME="E3:17_1_1_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</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#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E4:17_1_1_1_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</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#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, </i>;<br/><a NAME="E5:17_1_1_1_1_1_3_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, <a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_3_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</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#T7" target="_self">Th7</a>, , <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_1_1_1_3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, <a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, , <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_3_1_1_3"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/><a NAME="E3:17_1_1_1_1_1_3_1_1_3"/><b>then </b>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E4:17_1_1_1_1_1_3_1_1_3"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</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#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, </i>;<br/><a NAME="E5:17_1_1_1_1_1_3_1_1_3"/>
<font color="Maroon">x</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#T7" target="_self">Th7</a>, <a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_3_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:17_1_1_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_4"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_4"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1_1_1_4_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_4_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="xreal_1.html#T134">XREAL_1:134</a></i>;<br/><a NAME="E3:17_1_1_1_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</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#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E4:17_1_1_1_1_1_4_1_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</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#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, </i>;<br/><a NAME="E5:17_1_1_1_1_1_4_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, <a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_4_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_4_1_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</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#T7" target="_self">Th7</a>, , <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_1_1_1_4_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, <a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, , <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_4_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_4_1_1_3"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_4_1_1_3"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/><a NAME="E3:17_1_1_1_1_1_4_1_1_3"/><b>then </b>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E4:17_1_1_1_1_1_4_1_1_3"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</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#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, </i>;<br/><a NAME="E5:17_1_1_1_1_1_4_1_1_3"/>
<font color="Maroon">x</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#T7" target="_self">Th7</a>, <a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_4_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:17_1_1_1_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_5"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_5"/><i><font color="Green">E66</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_5"/><b>then </b><i><font color="Green">E67</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1_1_1_5_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_5_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_5_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/><a NAME="E3:17_1_1_1_1_1_5_1_1_1"/><b>then </b>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E4:17_1_1_1_1_1_5_1_1_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</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#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, </i>;<br/><a NAME="E5:17_1_1_1_1_1_5_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, <a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_5_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_5_1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_5_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</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#T7" target="_self">Th7</a>, , <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_1_1_1_5_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, <a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, , <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_5_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_5_1_1_3"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_5_1_1_3"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E8"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="xreal_1.html#T134">XREAL_1:134</a></i>;<br/><a NAME="E3:17_1_1_1_1_1_5_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</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#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E4:17_1_1_1_1_1_5_1_1_3"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</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#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, </i>;<br/><a NAME="E5:17_1_1_1_1_1_5_1_1_3"/>
<font color="Maroon">x</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#T7" target="_self">Th7</a>, <a class="txt" href="extreal1.html#E9"><i><font color="Green">Lemma22</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_5_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:17_1_1_1_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_6"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_6"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:17_1_1_1_1_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_7"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_7"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:17_1_1_1_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E4:17_1_1_1"><i><font color="Green">E20</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:17_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_2"/><i><font color="Green">E75</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_2_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> or  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> or <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  )
 </span><b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><b>hence </b><a NAME="E3:17_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_2"/><i><font color="Green">E76</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><a NAME="E3:17_1_1_2_1_1_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:17_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_3"/>
<font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_2_1_1_3"/><b>then </b>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><b>hence </b><a NAME="E3:17_1_1_2_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_3"/><i><font color="Green">E78</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_3_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> or  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> or <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  )
 </span><b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_3_1_1_1"/><i><font color="Green">E79</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E3:17_1_1_3_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , <a class="ref" href="extreal1.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_3_1_1_2"/><i><font color="Green">E81</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><a NAME="E3:17_1_1_3_1_1_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:17_1_1_3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_3_1_1_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_3_1_1_3"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E3:17_1_1_3_1_1_3"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</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#T7" target="_self">Th7</a>, , <a class="ref" href="extreal1.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_3_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_4"><b>suppose </b></a><a NAME="E1:17_1_1_4"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_4_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> or  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> or <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  )
 </span><b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_4_1_1_1"/><i><font color="Green">E86</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><b>hence </b><a NAME="E3:17_1_1_4_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_4_1_1_2"/><i><font color="Green">E87</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_4_1_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><a NAME="E3:17_1_1_4_1_1_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:17_1_1_4_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_4_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_4_1_1_3"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_4_1_1_3"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E3:17_1_1_4_1_1_3"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , </i>;<br/><b>hence </b><a NAME="E4:17_1_1_4_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_5"><b>suppose </b></a><a NAME="E1:17_1_1_5"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_5_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> or  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> or <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  )
 </span><b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_5_1_1_1"/><i><font color="Green">E92</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_5_1_1_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E3:17_1_1_5_1_1_1"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</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#T7" target="_self">Th7</a>, , <a class="ref" href="extreal1.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_5_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_5_1_1_2"/><i><font color="Green">E94</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_5_1_1_2"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><a NAME="E3:17_1_1_5_1_1_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:17_1_1_5_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_5_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_5_1_1_3"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_5_1_1_3"/><b>then </b><i><font color="Green">E97</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E3:17_1_1_5_1_1_3"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , <a class="ref" href="extreal1.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_5_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_6"><b>suppose </b></a><a NAME="E1:17_1_1_6"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_6_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> or  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> or <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  )
 </span><b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_6_1_1_1"/><i><font color="Green">E99</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_6_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><b>hence </b><a NAME="E3:17_1_1_6_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_6_1_1_2"/><i><font color="Green">E100</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_6_1_1_2"/><b>then </b><i><font color="Green">E101</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><a NAME="E3:17_1_1_6_1_1_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:17_1_1_6_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_6_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_6_1_1_3"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_6_1_1_3"/><b>then </b><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E3:17_1_1_6_1_1_3"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , </i>;<br/><b>hence </b><a NAME="E4:17_1_1_6_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_7"><b>suppose </b></a><a NAME="E1:17_1_1_7"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_7_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> or  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> or <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  )
 </span><b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_7_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_7_1_1_1"/><i><font color="Green">E105</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_7_1_1_1"/><b>then </b><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E3:17_1_1_7_1_1_1"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</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#T7" target="_self">Th7</a>, , <a class="ref" href="extreal1.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_7_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_7_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_7_1_1_2"/><i><font color="Green">E107</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_7_1_1_2"/><b>then </b><i><font color="Green">E108</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><a NAME="E3:17_1_1_7_1_1_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:17_1_1_7_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_7_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_7_1_1_3"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_7_1_1_3"/><b>then </b><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E3:17_1_1_7_1_1_3"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , <a class="ref" href="extreal1.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_7_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_8"><b>suppose </b></a><a NAME="E1:17_1_1_8"/><i><font color="Green">E111</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_8_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> or  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> or <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  )
 </span><b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_8_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_8_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_8_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><b>hence </b><a NAME="E3:17_1_1_8_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_8_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_8_1_1_2"/><i><font color="Green">E112</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_8_1_1_2"/><b>then </b><i><font color="Green">E113</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><a NAME="E3:17_1_1_8_1_1_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:17_1_1_8_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_8_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_8_1_1_3"/>
<font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_8_1_1_3"/><b>then </b>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><b>hence </b><a NAME="E3:17_1_1_8_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_9"><b>suppose </b></a><a NAME="E1:17_1_1_9"/><i><font color="Green">E114</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_9_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_9_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_9_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> or  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> or <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  )
 </span><b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_9_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_9_1_1_1"/><i><font color="Green">E115</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_9_1_1_1"/><b>then </b><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E3:17_1_1_9_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , <a class="ref" href="extreal1.html#T4" target="_self">Th4</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_9_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_9_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_9_1_1_2"/><i><font color="Green">E117</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_9_1_1_2"/><b>then </b><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><a NAME="E3:17_1_1_9_1_1_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:17_1_1_9_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_9_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_9_1_1_3"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_9_1_1_3"/><b>then </b><i><font color="Green">E120</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E3:17_1_1_9_1_1_3"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</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#T7" target="_self">Th7</a>, , <a class="ref" href="extreal1.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_9_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_10"><b>suppose </b></a><a NAME="E1:17_1_1_10"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_10"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:17_1_1_10"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_11"><b>suppose </b></a><a NAME="E1:17_1_1_11"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:17_1_1_11"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:17"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font></span>)</span>
 ;<br/>


</div>
