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

<b>assume </b><a NAME="E1:20"/><i><font color="Green">E18</font></i>: 
( <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>  )
 ;<br/>

<b>given </b><font color="Maroon">y</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><a NAME="E2:20"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</font> <a href="extreal1.html#K2">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc1.html#K10">1.</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 or 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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><b>suppose </b></a><a NAME="E1:20_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:20_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="extreal1.html#K2">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:20_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="mesfunc1.html#D8">MESFUNC1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2"><b>suppose </b></a><a NAME="E1:20_1_2"/><i><font color="Green">E20</font></i>: 
0 <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="20_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_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="E1:20_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">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:20_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="E2:20_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="extreal2.html#E1:20"><i><font color="Green">E18</font></i></a>, , <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:20_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="extreal2.html#E1:20"><i><font color="Green">E18</font></i></a>, , <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:20_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_3"><b>suppose </b></a><a NAME="E1:20_1_3"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_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="E1:20_1_3_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">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_3_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="E2:20_1_3_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="extreal2.html#E1:20"><i><font color="Green">E18</font></i></a>, , <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_3_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:20_1_3_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="txt" href="extreal2.html#E1:20"><i><font color="Green">E18</font></i></a>, , <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:20_1_3"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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