<?xml version="1.0"?>
<div class="add">

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







<b>assume </b><b>that </b><br/><a NAME="E1:60"/><i><font color="Green">E18</font></i>: 
( not <!--HUMANRECFAILED-->( <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>  ) &amp; not <!--HUMANRECFAILED-->( <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>  ) )
 <b>and </b><br/><a NAME="E2:60"/><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E3:60"/><i><font color="Green">E20</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font>
 ;<br/>

<a NAME="E4:60"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:60"/><b>then </b><i><font color="Green">E24</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 
<b>by </b><i><a class="txt" href="extreal2.html#E10:60"><i><font color="Green">E39</font></i></a>, <a class="txt" href="extreal2.html#E11:60"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E6:60"/><b>then </b>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">w</font> )
 
;<br/>
<a NAME="E7:60"/><b>then </b>
(  <a href="supinf_2.html#K3">-</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_2.html#K3">-</a> <font color="Maroon">w</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i><a class="ref" href="supinf_2.html#T17">SUPINF_2:17</a></i>;<br/>
<a NAME="E8:60"/><b>then </b><i><font color="Green">E25</font></i>: 
(  <a href="supinf_2.html#K3">-</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_2.html#K3">-</a> <font color="Maroon">w</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="extreal1.html#T4">EXTREAL1:4</a></i>;<br/>
<a NAME="E9:60"/><i><font color="Green">E38</font></i>: 
(  <a href="supinf_2.html#K3">-</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> &amp;  <a href="supinf_2.html#K3">-</a> <font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w</font> )
 
<b>by </b><i><a class="txt" href="extreal2.html#E10:60"><i><font color="Green">E39</font></i></a>, <a class="txt" href="extreal2.html#E11:60"><i><font color="Green">E40</font></i></a>, <a class="txt" href="extreal2.html#E21"><i><font color="Green">Lemma32</font></i></a></i>;<br/>
<a NAME="E10:60"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">w</font>
 
<b>by </b><i><a class="txt" href="extreal2.html#E9:60"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/>
<a NAME="E11:60"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <font color="Maroon">z</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <font color="Maroon">w</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="extreal2.html#E9:60"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="extreal2.html#T20" target="_self">Th20</a>, <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/>
<a NAME="E12:60"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <font color="Maroon">z</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">w</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E13:60"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="extreal2.html#T23" target="_self">Th23</a>, <a class="ref" href="extreal2.html#T24" target="_self">Th24</a>, <a class="txt" href="extreal2.html#E21"><i><font color="Green">Lemma32</font></i></a></i>;<br/>


</div>
