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

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





<b>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  or <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  )
 </span><b>by </b><i>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><b>suppose </b></a><a NAME="E1:34_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:34_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:34_1_1"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2"><b>suppose </b></a><a NAME="E1:34_1_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_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="E1:34_1_2_1_1"/>
( ( <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> ) or  not <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  or  not  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_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="E1:34_1_2_1_1_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_1_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1_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="E1:34_1_2_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  or <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  or  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> )
 </span><b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:34_1_2_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_1_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="E1:34_1_2_1_1_1_1_1_1_1_1_1_1_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> )
 </span><b>by </b><i><a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, <a class="txt" href="extreal1.html#E1:34_1_2_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_1_1_1_1_1_1_1_1_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_1_2_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:34_1_2_1_1_1"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_1_1_1_1_1_1_1_1_1_2"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_1_2_1_1_1_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="extreal1.html#E1:34_1_2_1_1_1_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="extreal1.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:34_1_2_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><a NAME="E2:34_1_2_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:34_1_2_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_1_1_1_1_1_1_3"/><i><font color="Green">E33</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:34_1_2_1_1_1_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1_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="E1:34_1_2_1_1_1_1_1_1_1_1_3_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> )
 </span><b>by </b><i><a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_1_1_1_1_1_1_3_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_1_2_1_1_1_1_1_1_1_1_3_1_1_1"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_1_1_1_1_1_1_3_1_1_2"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><a NAME="E2:34_1_2_1_1_1_1_1_1_1_1_3_1_1_2"/><b>then </b>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:34_1_2_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/><b>hence </b><a NAME="E3:34_1_2_1_1_1_1_1_1_1_1_3_1_1_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="txt" href="extreal1.html#E1:34_1_2_1_1_1_1_1_1"><i><font color="Green">E30</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:34_1_2_1_1_1_1_1_1_1_1_3"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:34_1_2_1_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_1_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_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="E1:34_1_2_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:34_1_2_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:34_1_2_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:34_1_2_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_2_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="E1:34_1_2_1_1_1_1_1_2_1_1_2_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_1_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:34_1_2_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:34_1_2_1_1_1_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T21">SUPINF_1:21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_1_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_1_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:34_1_2_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">y</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:34_1_2_1_1_1_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, </i>;<br/><a NAME="E5:34_1_2_1_1_1_1_1_2_1_1_2_1_1_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font> &amp; <font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="txt" href="extreal1.html#E1:34_1_2_1_1_1"><i><font color="Green">E29</font></i></a>, </i>;<br/><a NAME="E6:34_1_2_1_1_1_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><a NAME="E7:34_1_2_1_1_1_1_1_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:34_1_2_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xreal_1.html#T67">XREAL_1:67</a></i>;<br/><b>hence </b><a NAME="E8:34_1_2_1_1_1_1_1_2_1_1_2_1_1_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:34_1_2_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:34_1_2_1_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:34_1_2_1_1_1"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_2"/><i><font color="Green">E38</font></i>: 
(  not <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  or  not  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_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="E1:34_1_2_1_1_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 </span><b>by </b><i>, <a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_1_2_1_1_2_1_1_1"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_2_1_1_2_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_1_2_1_1_2_1_1_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T8" target="_self">Th8</a>, <a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:34_1_2_1_1_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:34_1_2"/>
<font color="Maroon">y</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="extreal1.html#K2" target="_self">*</a> <font color="Maroon">z</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
