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





<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green">E18</font></i>: 
(  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green">E19</font></i>: 
(  not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or  not <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="E3:30"/><i><font color="Green">E20</font></i>: 
(  not <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>and </b><br/><a NAME="E4:30"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font>
 ;<br/>

<b>thus </b><a NAME="E5:30"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/>

<a NAME="E2:30_1"/><b>then </b>
<font color="Maroon">z</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<b>by </b><i><a class="ref" href="extreal2.html#T7" target="_self">Th7</a>, <a class="ref" href="supinf_2.html#T6">SUPINF_2:6</a></i>;<br/>
<b>hence </b><a NAME="E3:30_1"/>
contradiction
 <b>by </b><i><a class="ref" href="extreal2.html#T4" target="_self">Th4</a>, <a class="ref" href="extreal2.html#T9" target="_self">Th9</a>, <a class="txt" href="extreal2.html#E5:30"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xxreal_0.html#T6">XXREAL_0:6</a></i>;<br/>


</div><b>end;</b></div>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_2"/>
( <font color="Maroon">y</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#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 </span><b>by </b><i><a class="txt" href="extreal2.html#E4:30"><i><font color="Green">E24</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1"><b>suppose </b></a><a NAME="E1:30_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:30_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E3:30"><i><font color="Green">E20</font></i></a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E3:30_2_1"/>
<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>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T5">XXREAL_0:5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_2"><b>suppose </b></a><a NAME="E1:30_2_2"/><i><font color="Green">E39</font></i>: 
( <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">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><a NAME="E2:30_2_2"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">y</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="extreal1.html#T1">EXTREAL1:1</a></i>;<br/><a NAME="E3:30_2_2"/>
(  <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a></i>;<br/><a NAME="E4:30_2_2"/><b>then </b><i><font color="Green">E41</font></i>: 
(  <a href="supinf_2.html#K3">-</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_2.html#K3">-</a> <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="txt" href="extreal2.html#E5:30"><i><font color="Green">E25</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> = 
<font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K3">-</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="txt" href="extreal2.html#E5:30"><i><font color="Green">E25</font></i></a>, <a class="txt" href="extreal2.html#E14"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="extreal1.html#T8">EXTREAL1:8</a></i>
<br/>.= 
<font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i><a class="ref" href="extreal1.html#T9">EXTREAL1:9</a></i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/><b>hence </b><a NAME="E6:30_2_2"/>
<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>
 <b>by </b><i><a class="ref" href="extreal2.html#T9" target="_self">Th9</a>, , <a class="ref" href="measure6.html#T11">MEASURE6:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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