<?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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <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="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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K2" title="SUPINF_1:func.2">-infty</a>  implies <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">y</font>
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green" title="E21">A2</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>  &amp; <font color="Maroon" title="c3">z</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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font></span><br/>

<a NAME="E3:42"/><i><font color="Green" title="E22">A3</font></i>: 
( <font color="Maroon" title="c1">x</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#E1:42"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T4" title="XXREAL_0:th.4">XXREAL_0:4</a>, <a class="ref" href="xxreal_0.html#T5" title="XXREAL_0:th.5">XXREAL_0:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">c</font> = <font color="Maroon" title="c3">z</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="extreal1.html#E2:42"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1"/>
( <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>  or <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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><b>suppose </b></a><a NAME="E1:42_1_1"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c2">y</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="extreal1.html#E3:42"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><a NAME="E3:42_1_1"/><i><font color="Green" title="E23">A4</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:42"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="extreal1.html#T32" target="_self" title="EXTREAL1:th.32">Th32</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_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="E1:42_1_1_1_1"/>
( <font color="Maroon" title="c1">x</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>  or <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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_1_1_1"/>
<font color="Maroon" title="c1">x</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c1">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="extreal1.html#E3:42"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><a NAME="E3:42_1_1_1_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:42"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="extreal1.html#T32" target="_self" title="EXTREAL1:th.32">Th32</a></i>;<br/><b>hence </b><a NAME="E4:42_1_1_1_1_1"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E1:42"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="extreal1.html#E2:42"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="extreal1.html#E3:42_1_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="real_1.html#T74" title="REAL_1:th.74">REAL_1:74</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_1_1_2"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><a NAME="E2:42_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c1">x</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</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#E2:42"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="extreal1.html#D2" target="_self" title="EXTREAL1:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E3:42_1_1_1_1_2"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:42_1_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T9" title="XXREAL_0:th.9">XXREAL_0:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:42_1_1"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2"><b>suppose </b></a><a NAME="E1:42_1_2"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><a NAME="E2:42_1_2"/><b>then </b><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</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#E2:42"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="extreal1.html#D2" target="_self" title="EXTREAL1:def.2">Def2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_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="E1:42_1_2_1_1"/>
( <font color="Maroon" title="c1">x</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>  or <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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_2_1_1_1"/>
<font color="Maroon" title="c1">x</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c1">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="extreal1.html#E3:42"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><a NAME="E3:42_1_2_1_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:42"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="extreal1.html#T32" target="_self" title="EXTREAL1:th.32">Th32</a></i>;<br/><b>hence </b><a NAME="E4:42_1_2_1_1_1"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:42_1_2"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T12" title="XXREAL_0:th.12">XXREAL_0:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_2_1_1_2"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font></span><br/><div class="add"><b>hence </b><a NAME="E2:42_1_2_1_1_2"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:42"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="extreal1.html#E2:42_1_2"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="extreal1.html#D2" target="_self" title="EXTREAL1:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:42_1_2"/>
<font color="Maroon" title="c2">y</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">x</font> <a href="extreal1.html#K3" title="EXTREAL1:func.3">/</a> <font color="Maroon" title="c3">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
