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<div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>;<br/>



<b>assume </b><a NAME="E1:14"/><i><font color="Green" title="E17">A1</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;  <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> ) 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="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><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_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> &amp;  <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> ) 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="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="txt" href="extreal1.html#E1:14"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1"/><i><font color="Green" title="E18">A2</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;  <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> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_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="E1:14_1_1_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> ) 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>  ) 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>  ) )
 </span><b>by </b><i><a class="txt" href="extreal1.html#E1:14_1_1_1"><i><font color="Green" title="E18">A2</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#T19" title="SUPINF_2:th.19">SUPINF_2:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_1_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="c3">a</font>, <font color="Maroon" title="c4">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:14_1_1_1_1_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c3">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">b</font> )
 ;<br/><a NAME="E4:14_1_1_1_1_1_1"/>
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">a</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="txt" href="extreal1.html#E1:14_1_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="extreal1.html#E3:14_1_1_1_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/><a NAME="E5:14_1_1_1_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a></i>;<br/><b>hence </b><a NAME="E6:14_1_1_1_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> <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#E3:14_1_1_1_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green" title="E10">Lm3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_1_1_1_2"/>
( ( (  <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>  ) 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>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_1_1_1_1_1_2"/>
 <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="ref" href="supinf_2.html#T19" title="SUPINF_2:th.19">SUPINF_2:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:14_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> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">y</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_2"/><i><font color="Green" title="E18">A4</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="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"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_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="E1:14_1_1_2_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 ( ( ( <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>  ) 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>  ) )
 </span><b>by </b><i><a class="txt" href="extreal1.html#E1:14_1_1_2"><i><font color="Green" title="E18">A4</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#T19" title="SUPINF_2:th.19">SUPINF_2:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:14_1_1_2_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="c3">a</font>, <font color="Maroon" title="c4">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:14_1_1_2_1_1_1"/><i><font color="Green" title="E19">A5</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c3">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">b</font> )
 ;<br/><a NAME="E4:14_1_1_2_1_1_1"/>
( <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 &amp; <font color="Maroon" title="c4">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#E1:14_1_1_2"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="extreal1.html#E3:14_1_1_2_1_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="supinf_2.html#D1" title="SUPINF_2:def.1">SUPINF_2:def 1</a></i>;<br/><a NAME="E5:14_1_1_2_1_1_1"/><b>then </b>
0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="ref" href="real_2.html#T122" title="REAL_2:th.122">REAL_2:122</a></i>;<br/><b>hence </b><a NAME="E6:14_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> <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#E3:14_1_1_2_1_1_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="extreal1.html#E17"><i><font color="Green" title="E10">Lm3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:14_1_1_2_1_1_2"/>
( ( ( <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>  ) 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>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_1_1_2_1_1_2"/>
 <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="ref" href="supinf_2.html#T19" title="SUPINF_2:th.19">SUPINF_2:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:14_1_1_2"/>
 <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>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:14"/>
 <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>
 ;<br/>


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