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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>assume </b><a NAME="E1:20"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<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"/>
( (  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp;  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) or  not  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or  not  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><b>case </b></a><b>that </b><a NAME="E1:20_1_1"/><i><font color="Green">E28</font></i>: 
 <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 <b>and </b><br/><a NAME="E2:20_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/><div class="add"><a NAME="E3:20_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:20_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E6:20_1_1"/><i><font color="Green">E50</font></i>: 
 <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E7:20_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:20_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E10:20_1_1"/><i><font color="Green">E53</font></i>: 
 <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/><b>take </b>
 <a href="real_1.html#K1">-</a> <font color="Maroon">b</font>
;<br/><b>take </b>
 <a href="real_1.html#K1">-</a> <font color="Maroon">a</font>
;<br/><b>thus </b><a NAME="E11:20_1_1"/>
(  <a href="real_1.html#K1">-</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">y</font> &amp;  <a href="real_1.html#K1">-</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="supinf_2.html#T2" target="_self">Th2</a>, <a class="txt" href="supinf_2.html#E1:20_1_1"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E12:20_1_1"/>
 ex <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">q</font> )
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T1" target="_self">Th1</a>, <a class="ref" href="supinf_2.html#D3" target="_self">Def3</a></i>;<br/><b>hence </b><a NAME="E13:20_1_1"/>
 <a href="real_1.html#K1">-</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="supinf_2.html#E1:20"><i><font color="Green">E24</font></i></a>, , <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2"><b>case </b></a><a NAME="E1:20_1_2"/>
(  not  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or  not  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/><div class="add"><a NAME="E2:20_1_2"/><b>then </b>
(  not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i/>;<br/><a NAME="E3:20_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i>, <a class="ref" href="supinf_1.html#D7">SUPINF_1:def 7</a></i>;<br/><b>hence </b><a NAME="E4:20_1_2"/>
(  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  <a href="supinf_2.html#K3" target="_self">-</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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