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<b>let </b><font color="Maroon">x</font> be   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">y</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:9"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/>

<a NAME="E2:9"/><b>then </b><i><font color="Green">E20</font></i>: 
(  <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>  &amp;  <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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1"/>
(  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">x</font> <a href="extreal1.html#K1" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="real_1.html#K4">*</a> <font color="Olive">b</font> ) or ( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) &amp; <font color="Maroon">x</font> <a href="extreal1.html#K1" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">x</font> <a href="extreal1.html#K1" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="real_1.html#K4">*</a> <font color="Olive">b</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:9_1_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">x</font> <a href="extreal1.html#K1" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b1</font> )
 ;<br/><b>thus </b><a NAME="E4:9_1_1_1"/>
<font color="Maroon">x</font> <a href="extreal1.html#K1" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_2"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  ) &amp; <font color="Maroon">x</font> <a href="extreal1.html#K1" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K1">0.</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_1_1_2"/>
<font color="Maroon">x</font> <a href="extreal1.html#K1" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="supinf_2.html#D1">SUPINF_2: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="E4:9"/>
<font color="Maroon">x</font> <a href="extreal1.html#K1" target="_self">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font>
 ;<br/>


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