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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="E1:50"/><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>  ) )
 ;<br/>

<a NAME="E2:50"/><i><font color="Green">E19</font></i>: 
(  <a href="supinf_2.html#K3">-</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> <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> <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;  <a href="supinf_2.html#K3">-</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> <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> <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><a class="txt" href="extreal2.html#E17"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1"/>
( <font color="Maroon">x</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>  or ( <font color="Maroon">x</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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><b>suppose </b></a><a NAME="E1:50_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:50_1_1"/><b>then </b>
(  <a href="supinf_1.html#K1">+infty</a>  <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; <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_1.html#K2">-infty</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="extreal2.html#T15" target="_self">Th15</a>, <a class="txt" href="extreal2.html#E17"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="xxreal_0.html#T5">XXREAL_0:5</a></i>;<br/><a NAME="E3:50_1_1"/><b>then </b><i><font color="Green">E24</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="hidden.html#R1">=</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> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_1_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+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>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:50_1_1_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:50_1_1_1_1_1"/><b>then </b>
<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="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T17" target="_self">Th17</a>, <a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/><a NAME="E3:50_1_1_1_1_1"/><b>then </b>
 <a href="supinf_2.html#K3">-</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="supinf_2.html#D3">SUPINF_2:def 3</a></i>;<br/><a NAME="E4:50_1_1_1_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span></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="ref" href="extreal2.html#T17" target="_self">Th17</a>, <a class="ref" href="extreal2.html#T18" target="_self">Th18</a>, <a class="txt" href="extreal2.html#E4:50_1_1_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/><a NAME="E5:50_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span></span>)</span> <a href="supinf_2.html#K4">-</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> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K2">+</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:50_1_1_1_1_1"/>
<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> <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="supinf_2.html#K2">+</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><a class="ref" href="extreal2.html#T18" target="_self">Th18</a>, <a class="ref" href="extreal2.html#T19" target="_self">Th19</a>, <a class="txt" href="extreal2.html#E4:50_1_1_1_1_1"><i><font color="Green">E38</font></i></a>, , <a class="txt" href="extreal2.html#E21"><i><font color="Green">Lemma32</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:50_1_1_1_1_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> 
 ;<br/><div class="add"><a NAME="E2:50_1_1_1_1_2"/><b>then </b><i><font color="Green">E40</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> <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="supinf_2.html#K2">+</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><a class="ref" href="extreal2.html#T17" target="_self">Th17</a>, <a class="ref" href="extreal2.html#T19" target="_self">Th19</a>, <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/><a NAME="E3:50_1_1_1_1_2"/><i><font color="Green">E41</font></i>: 
 <a href="supinf_2.html#K3">-</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T19" target="_self">Th19</a>, <a class="ref" href="supinf_2.html#D3">SUPINF_2:def 3</a></i>;<br/><a NAME="E4:50_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/><a NAME="E5:50_1_1_1_1_2"/><b>then </b>
 <a href="supinf_2.html#K3">-</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T17" target="_self">Th17</a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E6:50_1_1_1_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span></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="ref" href="extreal2.html#T15" target="_self">Th15</a>, <a class="ref" href="extreal2.html#T17" target="_self">Th17</a>, <a class="txt" href="extreal2.html#E2:50_1_1_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/><a NAME="E7:50_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span></span>)</span> <a href="supinf_2.html#K4">-</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> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K2">+</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:50_1_1_1_1_2"/>
<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> <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="supinf_2.html#K2">+</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><a class="txt" href="extreal2.html#E1:50_1_1_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="extreal2.html#E3:50_1_1_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="extreal2.html#E21"><i><font color="Green">Lemma32</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:50_1_1"/>
<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> <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="supinf_2.html#K2">+</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>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2"><b>suppose </b></a><a NAME="E1:50_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:50_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="supinf_2.html#K3">-</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> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T17" target="_self">Th17</a>, <a class="ref" href="xxreal_0.html#T6">XXREAL_0:6</a></i>;<br/><a NAME="E3:50_1_2"/><i><font color="Green">E53</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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="extreal2.html#T17" target="_self">Th17</a>, <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_2_1_1"/>
( <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#K2">-infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_1"/><i><font color="Green">E54</font></i>: 
<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="E2:50_1_2_1_1_1"/><b>then </b>
 <a href="supinf_2.html#K3">-</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> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal2.html#E17"><i><font color="Green">Lemma29</font></i></a></i>;<br/><a NAME="E3:50_1_2_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="supinf_2.html#K3">-</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> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="ref" href="xxreal_0.html#T6">XXREAL_0:6</a></i>;<br/><a NAME="E4:50_1_2_1_1_1"/><i><font color="Green">E56</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> <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="supinf_2.html#K2">+</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><a class="ref" href="extreal2.html#T17" target="_self">Th17</a>, <a class="ref" href="extreal2.html#T18" target="_self">Th18</a>, , <a class="ref" href="supinf_2.html#T4">SUPINF_2:4</a>, <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/><a NAME="E5:50_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span></span>)</span> <a href="supinf_2.html#K4">-</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> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K2">+</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:50_1_2_1_1_1"/>
<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> <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="supinf_2.html#K2">+</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><a class="ref" href="extreal2.html#T18" target="_self">Th18</a>, , , , <a class="txt" href="extreal2.html#E21"><i><font color="Green">Lemma32</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:50_1_2_1_1_2"/><i><font color="Green">E57</font></i>: 
<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="E2:50_1_2_1_1_2"/>
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T5">XXREAL_0:5</a></i>;<br/><a NAME="E3:50_1_2_1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:50_1_2_1_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&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><a class="ref" href="extreal2.html#T17" target="_self">Th17</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:50_1_2_1_1_2"/><i><font color="Green">E73</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> <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="supinf_2.html#K2">+</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><a class="ref" href="extreal2.html#T15" target="_self">Th15</a>, <a class="ref" href="extreal2.html#T17" target="_self">Th17</a>, , , <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/><a NAME="E6:50_1_2_1_1_2"/>
 <a href="supinf_2.html#K3">-</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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>, <a class="ref" href="supinf_2.html#T17">SUPINF_2:17</a></i>;<br/><a NAME="E7:50_1_2_1_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span></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="ref" href="extreal2.html#T15" target="_self">Th15</a>, <a class="ref" href="extreal2.html#T17" target="_self">Th17</a>, <a class="ref" href="extreal2.html#T19" target="_self">Th19</a>, <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/><a NAME="E8:50_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span></span>)</span> <a href="supinf_2.html#K4">-</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> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K2">+</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:50_1_2_1_1_2"/>
<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> <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="supinf_2.html#K2">+</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>, <a class="ref" href="extreal2.html#T20" target="_self">Th20</a>, <a class="txt" href="extreal2.html#E21"><i><font color="Green">Lemma32</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:50_1_2"/>
<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> <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="supinf_2.html#K2">+</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>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_3"><b>suppose </b></a><a NAME="E1:50_1_3"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">x</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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/><div class="add"><a NAME="E2:50_1_3"/><b>then </b>
(  <a href="supinf_2.html#K3">-</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_2.html#K3">-</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>by </b><i><a class="ref" href="extreal1.html#T4">EXTREAL1:4</a></i>;<br/><a NAME="E3:50_1_3"/><b>then </b><i><font color="Green">E76</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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>by </b><i><a class="ref" href="extreal2.html#T18" target="_self">Th18</a>, </i>;<br/><a NAME="E4:50_1_3"/><b>then </b><i><font color="Green">E77</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> <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="supinf_2.html#K2">+</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><a class="ref" href="extreal2.html#T17" target="_self">Th17</a>, <a class="ref" href="extreal2.html#T18" target="_self">Th18</a>, <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/><a NAME="E5:50_1_3"/>
(  <a href="supinf_2.html#K3">-</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_2.html#K3">-</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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 <b>by </b><i><a class="ref" href="extreal2.html#T19" target="_self">Th19</a>, <a class="ref" href="extreal1.html#T4">EXTREAL1:4</a></i>;<br/><a NAME="E6:50_1_3"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span></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="ref" href="extreal2.html#T17" target="_self">Th17</a>, <a class="ref" href="extreal2.html#T18" target="_self">Th18</a>, <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/><a NAME="E7:50_1_3"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span></span>)</span> <a href="supinf_2.html#K4">-</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> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">x</font></span><a href="extreal1.html#K4">.|</a></span> <a href="supinf_2.html#K2">+</a> <span class="p2"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">y</font></span><a href="extreal1.html#K4">.|</a></span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:50_1_3"/>
<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> <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="supinf_2.html#K2">+</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>, , <a class="txt" href="extreal2.html#E21"><i><font color="Green">Lemma32</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
