<?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>, <font color="Maroon">y</font> be   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">s</font> be   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">t</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:17"/><i><font color="Green">E24</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">s</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">s</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> <a href="supinf_2.html#K2" target="_self">+</a> <font color="Maroon">s</font>, <font color="Maroon">b</font> = <font color="Maroon">y</font> <a href="supinf_2.html#K2" target="_self">+</a> <font color="Maroon">t</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E3:17"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K2">}</a></span> ) &amp; ( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K2">}</a></span> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:17"/><b>then </b><i><font color="Green">E28</font></i>: 
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:17"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  implies <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/>

<a NAME="E2:17_1"/><b>then </b><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">y</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">s</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">t</font> is   <a href="real_1.html#NM1">Real</a> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:17_1"/><b>then </b><i><font color="Green">E31</font></i>: 
 ex <font color="Olive">Ox</font>, <font color="Olive">Oy</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">Ox</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">Oy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Olive">Ox</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">Oy</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:17_1"/>
 ex <font color="Olive">Os</font>, <font color="Olive">Ot</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">Os</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Olive">Ot</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <font color="Olive">Os</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">Ot</font> )
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">Ox</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">Oy</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">Os</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">Ot</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:17_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">Ox</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">Oy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">Os</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">Ot</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">Ox</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Oy</font> &amp; <font color="Maroon">Os</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Ot</font> )
 <b>by </b><i/>;<br/>
<a NAME="E7:17_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">Ox</font> <a href="real_1.html#K3">+</a> <font color="Maroon">Os</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Os</font> <a href="real_1.html#K3">+</a> <font color="Maroon">Oy</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T1" target="_self">Th1</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:17_1"/>
<font color="Maroon">Os</font> <a href="real_1.html#K3">+</a> <font color="Maroon">Oy</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Ot</font> <a href="real_1.html#K3">+</a> <font color="Maroon">Oy</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T1" target="_self">Th1</a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E9:17_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">Ox</font> <a href="real_1.html#K3">+</a> <font color="Maroon">Os</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Oy</font> <a href="real_1.html#K3">+</a> <font color="Maroon">Ot</font>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:17"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E10:17_1"/>
( <font color="Maroon">x</font> <a href="supinf_2.html#K2" target="_self">+</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ox</font> <a href="real_1.html#K3">+</a> <font color="Maroon">Os</font> &amp; <font color="Maroon">y</font> <a href="supinf_2.html#K2" target="_self">+</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Oy</font> <a href="real_1.html#K3">+</a> <font color="Maroon">Ot</font> )
 
<b>by </b><i><a class="ref" href="supinf_2.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E11:17_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:17"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  implies <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/>

<a NAME="E2:17_2"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E3:17_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">s</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a></i>;<br/>
<a NAME="E4:17_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:17_2"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T21">SUPINF_1:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:17"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  implies <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_3"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/>

<a NAME="E2:17_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E3:17_3"/><b>then </b>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">t</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#T24">SUPINF_1:24</a></i>;<br/>
<a NAME="E4:17_3"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:17_3"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:17"/>
(  not <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or  not <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_4"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/>

<a NAME="E2:17_4"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E3:17_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">s</font> <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#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E4:17_4"/>
contradiction
 <b>by </b><i>, <a class="txt" href="supinf_2.html#E1:17"><i><font color="Green">E24</font></i></a>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:17"/>
<font color="Maroon">x</font> <a href="supinf_2.html#K2" target="_self">+</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2" target="_self">+</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="supinf_2.html#D2" target="_self">Def2</a>, , , , <a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a>, <a class="ref" href="supinf_1.html#T21">SUPINF_1:21</a></i>;<br/>


</div>
