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<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><a NAME="E1:35"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">x</font> <a href="extreal1.html#K2">*</a> <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">x</font> <a href="extreal1.html#K2">*</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/>

<b>assume </b><a NAME="E2:35"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">x</font> is  not   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">y</font> is  not   <a href="real_1.html#NM1">Real</a> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">y</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#K1">+infty</a>  &amp; <font color="Maroon">y</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#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">y</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>  &amp; <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="extreal2.html#T7" target="_self">Th7</a>, <a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>suppose </b></a><a NAME="E1:35_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:35_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="extreal2.html#T4" target="_self">Th4</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="35_1_2"><b>suppose </b></a><a NAME="E1:35_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <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="E2:35_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="extreal2.html#T4" target="_self">Th4</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="35_1_3"><b>suppose </b></a><a NAME="E1:35_1_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:35_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="extreal2.html#T4" target="_self">Th4</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="35_1_4"><b>suppose </b></a><a NAME="E1:35_1_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <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="E2:35_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="extreal2.html#T4" target="_self">Th4</a>, <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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