<?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>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green">E18</font></i>: 
( ( not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or ( not <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; not <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <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#K1">+infty</a>  &amp; <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="54_1_1"><b>suppose </b></a><a NAME="E1:54_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:54_1_1"/><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="extreal2.html#T17" target="_self">Th17</a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><a NAME="E3:54_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/><a NAME="E4:54_1_1"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T19" target="_self">Th19</a>, <a class="ref" href="extreal1.html#T36">EXTREAL1:36</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_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="E1:54_1_1_1_1"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span><b>by </b><i><a class="ref" href="extreal2.html#T18" target="_self">Th18</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:54_1_1_1_1_1"/><i><font color="Green">E38</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><a NAME="E2:54_1_1_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T36">EXTREAL1:36</a></i>;<br/><a NAME="E3:54_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T19" target="_self">Th19</a>, <a class="txt" href="extreal2.html#E1:54_1_1_1_1_1"><i><font color="Green">E38</font></i></a>, , </i>;<br/><a NAME="E4:54_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T36">EXTREAL1:36</a></i>;<br/><b>hence </b><a NAME="E5:54_1_1_1_1_1"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="extreal1.html#K3">/</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i><a class="txt" href="extreal2.html#E1:54_1_1_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="extreal2.html#E2:54_1_1_1_1_1"><i><font color="Green">E39</font></i></a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:54_1_1_1_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:54_1_1_1_1_2"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T37">EXTREAL1:37</a></i>;<br/><a NAME="E3:54_1_1_1_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a>, <a class="ref" href="supinf_2.html#T17">SUPINF_2:17</a></i>;<br/><a NAME="E4:54_1_1_1_1_2"/><i><font color="Green">E42</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i><a class="ref" href="extreal2.html#T18" target="_self">Th18</a>, <a class="txt" href="extreal2.html#E14"><i><font color="Green">Lemma27</font></i></a></i>;<br/><a NAME="E5:54_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T19" target="_self">Th19</a>, , <a class="txt" href="extreal2.html#E1:54_1_1_1_1_2"><i><font color="Green">E40</font></i></a>, </i>;<br/><a NAME="E6:54_1_1_1_1_2"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T37">EXTREAL1:37</a>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a></i>;<br/><b>hence </b><a NAME="E7:54_1_1_1_1_2"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="extreal1.html#K3">/</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i>, <a class="txt" href="extreal2.html#E3:54_1_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="extreal2.html#E4:54_1_1_1_1_2"><i><font color="Green">E42</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:54_1_1"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="extreal1.html#K3">/</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2"><b>suppose </b></a><a NAME="E1:54_1_2"/><i><font color="Green">E50</font></i>: 
<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="E2:54_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="extreal2.html#T17" target="_self">Th17</a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><a NAME="E3:54_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/><a NAME="E4:54_1_2"/><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T19" target="_self">Th19</a>, <a class="ref" href="extreal1.html#T37">EXTREAL1:37</a>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_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="E1:54_1_2_1_1"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span><b>by </b><i><a class="ref" href="extreal2.html#T18" target="_self">Th18</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:54_1_2_1_1_1"/><i><font color="Green">E54</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 ;<br/><div class="add"><a NAME="E2:54_1_2_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T36">EXTREAL1:36</a></i>;<br/><a NAME="E3:54_1_2_1_1_1"/>
<font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T19" target="_self">Th19</a>, , , </i>;<br/><a NAME="E4:54_1_2_1_1_1"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T37">EXTREAL1:37</a>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a></i>;<br/><b>hence </b><a NAME="E5:54_1_2_1_1_1"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="extreal1.html#K3">/</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:54_1_2_1_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:54_1_2_1_1_2"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="extreal1.html#T37">EXTREAL1:37</a></i>;<br/><a NAME="E3:54_1_2_1_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a>, <a class="ref" href="supinf_2.html#T17">SUPINF_2:17</a></i>;<br/><a NAME="E4:54_1_2_1_1_2"/><i><font color="Green">E71</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i><a class="ref" href="extreal2.html#T18" target="_self">Th18</a>, <a class="txt" href="extreal2.html#E14"><i><font color="Green">Lemma27</font></i></a></i>;<br/><a NAME="E5:54_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T19" target="_self">Th19</a>, , , </i>;<br/><a NAME="E6:54_1_2_1_1_2"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal1.html#T36">EXTREAL1:36</a></i>;<br/><b>hence </b><a NAME="E7:54_1_2_1_1_2"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="extreal1.html#K3">/</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:54_1_2"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="extreal1.html#K3">/</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_3"><b>suppose </b></a><a NAME="E1:54_1_3"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <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"><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><a NAME="E3:54_1_3"/>
<font color="Maroon">x</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="extreal2.html#T19" target="_self">Th19</a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><a NAME="E4:54_1_3"/><b>then </b>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/><a NAME="E5:54_1_3"/><b>then </b>
(  <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="extreal1.html#T4">EXTREAL1:4</a></i>;<br/><a NAME="E6:54_1_3"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E20"><i><font color="Green">Lemma31</font></i></a></i>;<br/><a NAME="E7:54_1_3"/><i><font color="Green">E74</font></i>: 
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_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="E1:54_1_3_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#R1">=</a>  <a href="supinf_1.html#K2">-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>  &amp; <font color="Maroon">y</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="54_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:54_1_3_1_1_1"/><i><font color="Green">E75</font></i>: 
<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="E2:54_1_3_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="extreal2.html#T19" target="_self">Th19</a>, <a class="ref" href="extreal1.html#T33">EXTREAL1:33</a></i>;<br/><a NAME="E3:54_1_3_1_1_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="extreal1.html#D3">EXTREAL1:def 3</a></i>;<br/><a NAME="E4:54_1_3_1_1_1"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="extreal1.html#T36">EXTREAL1:36</a></i>;<br/><b>hence </b><a NAME="E5:54_1_3_1_1_1"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="extreal1.html#K3">/</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i>, , , <a class="ref" href="extreal1.html#T33">EXTREAL1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:54_1_3_1_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:54_1_3_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="extreal2.html#T19" target="_self">Th19</a>, <a class="ref" href="extreal1.html#T33">EXTREAL1:33</a></i>;<br/><a NAME="E3:54_1_3_1_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="extreal1.html#D3">EXTREAL1:def 3</a></i>;<br/><a NAME="E4:54_1_3_1_1_2"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i>, <a class="ref" href="extreal1.html#T37">EXTREAL1:37</a>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a></i>;<br/><b>hence </b><a NAME="E5:54_1_3_1_1_2"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="extreal1.html#K3">/</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i>, , , <a class="ref" href="extreal1.html#T33">EXTREAL1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:54_1_3_1_1_3"/>
( <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">y</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><a NAME="E3:54_1_3_1_1_3"/>
<font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K6">/</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="extreal2.html#T18" target="_self">Th18</a>, <a class="ref" href="extreal1.html#T32">EXTREAL1:32</a></i>;<br/><a NAME="E4:54_1_3_1_1_3"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K6">/</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:54_1_3_1_1_3"/><i><font color="Green">E81</font></i>: 
0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i><a class="ref" href="extreal2.html#T18" target="_self">Th18</a>, <a class="txt" href="extreal2.html#E14"><i><font color="Green">Lemma27</font></i></a></i>;<br/><a NAME="E6:54_1_3_1_1_3"/>
( <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">a</font> &amp; <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/><a NAME="E7:54_1_3_1_1_3"/><b>then </b>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="extreal1.html#K3">/</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="extreal1.html#T32">EXTREAL1:32</a></i>;<br/><b>hence </b><a NAME="E8:54_1_3_1_1_3"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="extreal1.html#K3">/</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span>
 <b>by </b><i>, <a class="ref" href="complex1.html#T153">COMPLEX1:153</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:54_1_3"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="extreal1.html#K3">/</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="extreal1.html#K3">/</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
