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

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">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" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b2">b</font> ) or (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) or (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) or ( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) or ( <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) or ( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) or ( <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) or (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) or (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) or ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  ) or ( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  ) )
 </span><b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b2">b</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:17_1_1_1"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font> )
 ;<br/><a NAME="E4:17_1_1_1"/><i><font color="Green" title="E19">A2</font></i>: 
( <font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1_1_1"/>
(  ex <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font> &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">c</font> &amp; <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b2">c</font> ) or (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) or ( <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  ) or ( <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) or (  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  ) or ( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  ) or ( <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  ) )
 </span><b>by </b><i><a class="txt" href="extreal1.html#E4:17_1_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font> &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">c</font> &amp; <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Olive" title="b2">c</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">b1</font>, <font color="Maroon" title="c7">c</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:17_1_1_1_1_1_1"/><i><font color="Green" title="E20">A3</font></i>: 
( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> &amp; <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">c</font> )
 ;<br/><a NAME="E4:17_1_1_1_1_1_1"/><i><font color="Green" title="E21">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">c</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E3:17_1_1_1_1_1_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self" title="EXTREAL1:th.13">Th13</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c7">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E3:17_1_1_1_1_1_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="extreal1.html#T13" target="_self" title="EXTREAL1:th.13">Th13</a></i>;<br/><b>hence </b><a NAME="E6:17_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_1"><i><font color="Green" title="E21">A4</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" title="E20">A5</font></i>: 
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">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" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E22">A7</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_1_1_1_2"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="real_2.html#T122" title="REAL_2:th.122">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" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green" title="E10">Lm3</font></i></a></i>;<br/><a NAME="E4:17_1_1_1_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E23">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_2"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_2_1_1_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green" title="E10">Lm3</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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_2"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E22">A9</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">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" title="E22">A10</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_1_1_1_2"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="xreal_1.html#T134" title="XREAL_1:th.134">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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E18"><i><font color="Green" title="E11">Lm4</font></i></a></i>;<br/><a NAME="E4:17_1_1_1_1_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E23">A11</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_2"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_2_1_1_3"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_2_1_1_3"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="extreal1.html#E18"><i><font color="Green" title="E11">Lm4</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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_2"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_2_1_1_3"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:17_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">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" title="E20">A12</font></i>: 
( <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E21">A13</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">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" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E22">A14</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_1_1_1_3"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="xreal_1.html#T134" title="XREAL_1:th.134">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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E18"><i><font color="Green" title="E11">Lm4</font></i></a></i>;<br/><a NAME="E4:17_1_1_1_1_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E23">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_3"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_3_1_1_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E22">A14</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green" title="E10">Lm3</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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_3"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E23">A15</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_3_1_1_2"/><i><font color="Green" title="E22">A16</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E22">A16</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">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" title="E22">A17</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_1_1_1_3"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="real_2.html#T122" title="REAL_2:th.122">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" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green" title="E10">Lm3</font></i></a></i>;<br/><a NAME="E4:17_1_1_1_1_1_3_1_1_3"/><b>then </b><i><font color="Green" title="E23">A18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_3"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_3_1_1_3"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_3_1_1_3"><i><font color="Green" title="E22">A17</font></i></a>, <a class="txt" href="extreal1.html#E18"><i><font color="Green" title="E11">Lm4</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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_3"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_3_1_1_3"><i><font color="Green" title="E23">A18</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:17_1_1_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">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" title="E20">A19</font></i>: 
( <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_4"/><b>then </b><i><font color="Green" title="E21">A20</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">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" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E22">A21</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_1_1_1_4"><i><font color="Green" title="E21">A20</font></i></a>, <a class="ref" href="xreal_1.html#T134" title="XREAL_1:th.134">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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E18"><i><font color="Green" title="E11">Lm4</font></i></a></i>;<br/><a NAME="E4:17_1_1_1_1_1_4_1_1_1"/><b>then </b><i><font color="Green" title="E23">A22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_4"><i><font color="Green" title="E20">A19</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_4_1_1_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_4_1_1_1"><i><font color="Green" title="E22">A21</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green" title="E10">Lm3</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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_4"><i><font color="Green" title="E20">A19</font></i></a>, <a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_4_1_1_1"><i><font color="Green" title="E23">A22</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_4_1_1_2"/><i><font color="Green" title="E22">A23</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_4_1_1_2"><i><font color="Green" title="E22">A23</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">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" title="E22">A24</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_1_1_1_4"><i><font color="Green" title="E21">A20</font></i></a>, <a class="ref" href="real_2.html#T122" title="REAL_2:th.122">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" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green" title="E10">Lm3</font></i></a></i>;<br/><a NAME="E4:17_1_1_1_1_1_4_1_1_3"/><b>then </b><i><font color="Green" title="E23">A25</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_4"><i><font color="Green" title="E20">A19</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_4_1_1_3"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_4_1_1_3"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="extreal1.html#E18"><i><font color="Green" title="E11">Lm4</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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_4"><i><font color="Green" title="E20">A19</font></i></a>, <a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_4_1_1_3"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:17_1_1_1_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">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" title="E20">A26</font></i>: 
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  )
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1_5"/><b>then </b><i><font color="Green" title="E21">A27</font></i>: 
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">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" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="E22">A28</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&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" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_1_1_1_5"><i><font color="Green" title="E21">A27</font></i></a>, <a class="ref" href="real_2.html#T122" title="REAL_2:th.122">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" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green" title="E10">Lm3</font></i></a></i>;<br/><a NAME="E4:17_1_1_1_1_1_5_1_1_1"/><b>then </b><i><font color="Green" title="E23">A29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_5"><i><font color="Green" title="E20">A26</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_5_1_1_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_5_1_1_1"><i><font color="Green" title="E22">A28</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green" title="E10">Lm3</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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_5"><i><font color="Green" title="E20">A26</font></i></a>, <a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_5_1_1_1"><i><font color="Green" title="E23">A29</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_5_1_1_2"/><i><font color="Green" title="E22">A30</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_5_1_1_2"><i><font color="Green" title="E22">A30</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">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" title="E22">A31</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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" title="c4">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_1_1_1_5"><i><font color="Green" title="E21">A27</font></i></a>, <a class="ref" href="xreal_1.html#T134" title="XREAL_1:th.134">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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E18"><i><font color="Green" title="E11">Lm4</font></i></a></i>;<br/><a NAME="E4:17_1_1_1_1_1_5_1_1_3"/><b>then </b><i><font color="Green" title="E23">A32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_5"><i><font color="Green" title="E20">A26</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E5:17_1_1_1_1_1_5_1_1_3"/>
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E3:17_1_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_5_1_1_3"><i><font color="Green" title="E22">A31</font></i></a>, <a class="txt" href="extreal1.html#E18"><i><font color="Green" title="E11">Lm4</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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_5"><i><font color="Green" title="E20">A26</font></i></a>, <a class="txt" href="extreal1.html#E4:17_1_1_1_1_1_5_1_1_3"><i><font color="Green" title="E23">A32</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:17_1_1_1_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">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" title="E20">A33</font></i>: 
( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_1_1_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_6"><i><font color="Green" title="E20">A33</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1_7"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1_7"/><i><font color="Green" title="E20">A34</font></i>: 
( <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_1_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_1_1_1_7"><i><font color="Green" title="E20">A34</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:17_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">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" title="E18">A35</font></i>: 
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+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" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">z</font> or  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> or <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_2"><i><font color="Green" title="E18">A35</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_2"><i><font color="Green" title="E18">A35</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_2"/><i><font color="Green" title="E19">A36</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E20">A37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_2_1_1_2"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_2_1_1_2"><i><font color="Green" title="E19">A36</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_2_1_1_2"><i><font color="Green" title="E20">A37</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_3"/>
<font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_2_1_1_3"/><b>then </b>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_2"><i><font color="Green" title="E18">A35</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_2_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_2"><i><font color="Green" title="E18">A35</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">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" title="E18">A38</font></i>: 
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+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" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">z</font> or  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> or <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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" title="E19">A39</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E20">A40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_3"><i><font color="Green" title="E18">A38</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_3_1_1_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_3"><i><font color="Green" title="E18">A38</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_3_1_1_1"><i><font color="Green" title="E19">A39</font></i></a>, <a class="ref" href="extreal1.html#T20" target="_self" title="EXTREAL1:th.20">Th20</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_3"><i><font color="Green" title="E18">A38</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_3_1_1_1"><i><font color="Green" title="E20">A40</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_3_1_1_2"/><i><font color="Green" title="E19">A41</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E20">A42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_3_1_1_2"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_3_1_1_2"><i><font color="Green" title="E19">A41</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_3_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_3_1_1_2"><i><font color="Green" title="E20">A42</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_3_1_1_3"/><i><font color="Green" title="E19">A43</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_3_1_1_3"/><b>then </b><i><font color="Green" title="E20">A44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_3"><i><font color="Green" title="E18">A38</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_3_1_1_3"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_3"><i><font color="Green" title="E18">A38</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_3_1_1_3"><i><font color="Green" title="E19">A43</font></i></a>, <a class="ref" href="extreal1.html#T21" target="_self" title="EXTREAL1:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_3_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_3"><i><font color="Green" title="E18">A38</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_3_1_1_3"><i><font color="Green" title="E20">A44</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">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" title="E18">A45</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+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" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">z</font> or  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> or <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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" title="E19">A46</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_4"><i><font color="Green" title="E18">A45</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_4_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_4"><i><font color="Green" title="E18">A45</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_4_1_1_1"><i><font color="Green" title="E19">A46</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_4_1_1_2"/><i><font color="Green" title="E19">A47</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_4_1_1_2"/><b>then </b><i><font color="Green" title="E20">A48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_4_1_1_2"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_4_1_1_2"><i><font color="Green" title="E19">A47</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_4_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_4_1_1_2"><i><font color="Green" title="E20">A48</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_4_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_4_1_1_3"/><i><font color="Green" title="E19">A49</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_4_1_1_3"/><b>then </b><i><font color="Green" title="E20">A50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_4"><i><font color="Green" title="E18">A45</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_4_1_1_3"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_4"><i><font color="Green" title="E18">A45</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_4_1_1_3"><i><font color="Green" title="E19">A49</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_4_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_4"><i><font color="Green" title="E18">A45</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_4_1_1_3"><i><font color="Green" title="E20">A50</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">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" title="E18">A51</font></i>: 
( <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+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" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">z</font> or  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> or <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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" title="E19">A52</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_5_1_1_1"/><b>then </b><i><font color="Green" title="E20">A53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_5"><i><font color="Green" title="E18">A51</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_5_1_1_1"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_5"><i><font color="Green" title="E18">A51</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_5_1_1_1"><i><font color="Green" title="E19">A52</font></i></a>, <a class="ref" href="extreal1.html#T21" target="_self" title="EXTREAL1:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_5_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_5"><i><font color="Green" title="E18">A51</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_5_1_1_1"><i><font color="Green" title="E20">A53</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_5_1_1_2"/><i><font color="Green" title="E19">A54</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_5_1_1_2"/><b>then </b><i><font color="Green" title="E20">A55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_5_1_1_2"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_5_1_1_2"><i><font color="Green" title="E19">A54</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_5_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_5_1_1_2"><i><font color="Green" title="E20">A55</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_5_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_5_1_1_3"/><i><font color="Green" title="E19">A56</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_5_1_1_3"/><b>then </b><i><font color="Green" title="E20">A57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_5"><i><font color="Green" title="E18">A51</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_5_1_1_3"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_5"><i><font color="Green" title="E18">A51</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_5_1_1_3"><i><font color="Green" title="E19">A56</font></i></a>, <a class="ref" href="extreal1.html#T20" target="_self" title="EXTREAL1:th.20">Th20</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_5_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_5"><i><font color="Green" title="E18">A51</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_5_1_1_3"><i><font color="Green" title="E20">A57</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">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" title="E18">A58</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-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" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">z</font> or  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> or <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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" title="E19">A59</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_6_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_6"><i><font color="Green" title="E18">A58</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_6_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_6"><i><font color="Green" title="E18">A58</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_6_1_1_1"><i><font color="Green" title="E19">A59</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_6_1_1_2"/><i><font color="Green" title="E19">A60</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_6_1_1_2"/><b>then </b><i><font color="Green" title="E20">A61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_6_1_1_2"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_6_1_1_2"><i><font color="Green" title="E19">A60</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_6_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_6_1_1_2"><i><font color="Green" title="E20">A61</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_6_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_6_1_1_3"/><i><font color="Green" title="E19">A62</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_6_1_1_3"/><b>then </b><i><font color="Green" title="E20">A63</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_6"><i><font color="Green" title="E18">A58</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_6_1_1_3"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_6"><i><font color="Green" title="E18">A58</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_6_1_1_3"><i><font color="Green" title="E19">A62</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_6_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_6"><i><font color="Green" title="E18">A58</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_6_1_1_3"><i><font color="Green" title="E20">A63</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">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" title="E18">A64</font></i>: 
( <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-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" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">z</font> or  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> or <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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" title="E19">A65</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_7_1_1_1"/><b>then </b><i><font color="Green" title="E20">A66</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_7"><i><font color="Green" title="E18">A64</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_7_1_1_1"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_7"><i><font color="Green" title="E18">A64</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_7_1_1_1"><i><font color="Green" title="E19">A65</font></i></a>, <a class="ref" href="extreal1.html#T21" target="_self" title="EXTREAL1:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_7_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_7"><i><font color="Green" title="E18">A64</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_7_1_1_1"><i><font color="Green" title="E20">A66</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_7_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_7_1_1_2"/><i><font color="Green" title="E19">A67</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_7_1_1_2"/><b>then </b><i><font color="Green" title="E20">A68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_7_1_1_2"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_7_1_1_2"><i><font color="Green" title="E19">A67</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_7_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_7_1_1_2"><i><font color="Green" title="E20">A68</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_7_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_7_1_1_3"/><i><font color="Green" title="E19">A69</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_7_1_1_3"/><b>then </b><i><font color="Green" title="E20">A70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_7"><i><font color="Green" title="E18">A64</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_7_1_1_3"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_7"><i><font color="Green" title="E18">A64</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_7_1_1_3"><i><font color="Green" title="E19">A69</font></i></a>, <a class="ref" href="extreal1.html#T20" target="_self" title="EXTREAL1:th.20">Th20</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_7_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_7"><i><font color="Green" title="E18">A64</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_7_1_1_3"><i><font color="Green" title="E20">A70</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">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" title="E18">A71</font></i>: 
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-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" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">z</font> or  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> or <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_8_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_8"><i><font color="Green" title="E18">A71</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_8_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_8"><i><font color="Green" title="E18">A71</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_8_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_8_1_1_2"/><i><font color="Green" title="E19">A72</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_8_1_1_2"/><b>then </b><i><font color="Green" title="E20">A73</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_8_1_1_2"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_8_1_1_2"><i><font color="Green" title="E19">A72</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_8_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_8_1_1_2"><i><font color="Green" title="E20">A73</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_8_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_8_1_1_3"/>
<font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_8_1_1_3"/><b>then </b>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_8"><i><font color="Green" title="E18">A71</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_8_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_8"><i><font color="Green" title="E18">A71</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">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" title="E18">A74</font></i>: 
(  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-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" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">z</font> or  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> or <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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" title="E19">A75</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_9_1_1_1"/><b>then </b><i><font color="Green" title="E20">A76</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_9"><i><font color="Green" title="E18">A74</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_9_1_1_1"/>
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_9"><i><font color="Green" title="E18">A74</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_9_1_1_1"><i><font color="Green" title="E19">A75</font></i></a>, <a class="ref" href="extreal1.html#T20" target="_self" title="EXTREAL1:th.20">Th20</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_9_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_9"><i><font color="Green" title="E18">A74</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_9_1_1_1"><i><font color="Green" title="E20">A76</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_9_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_9_1_1_2"/><i><font color="Green" title="E19">A77</font></i>: 
 <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_9_1_1_2"/><b>then </b><i><font color="Green" title="E20">A78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_9_1_1_2"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_9_1_1_2"><i><font color="Green" title="E19">A77</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_9_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:17_1_1_9_1_1_2"><i><font color="Green" title="E20">A78</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_9_1_1_3"><b>suppose </b></a><a NAME="E1:17_1_1_9_1_1_3"/><i><font color="Green" title="E19">A79</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:17_1_1_9_1_1_3"/><b>then </b><i><font color="Green" title="E20">A80</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_9"><i><font color="Green" title="E18">A74</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><a NAME="E3:17_1_1_9_1_1_3"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_9"><i><font color="Green" title="E18">A74</font></i></a>, <a class="txt" href="extreal1.html#E1:17_1_1_9_1_1_3"><i><font color="Green" title="E19">A79</font></i></a>, <a class="ref" href="extreal1.html#T21" target="_self" title="EXTREAL1:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E4:17_1_1_9_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_9"><i><font color="Green" title="E18">A74</font></i></a>, <a class="txt" href="extreal1.html#E2:17_1_1_9_1_1_3"><i><font color="Green" title="E20">A80</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">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" title="E18">A81</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_10"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_10"><i><font color="Green" title="E18">A81</font></i></a>, <a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_11"><b>suppose </b></a><a NAME="E1:17_1_1_11"/><i><font color="Green" title="E18">A82</font></i>: 
( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">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" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K1" title="SUPINF_2:func.1">0.</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#D1" target="_self" title="EXTREAL1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_11"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:17_1_1_11"><i><font color="Green" title="E18">A82</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:17"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">z</font></span>)</span>
 ;<br/>


</div>
