<?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="E1:18"/><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">t</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+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">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#K1">+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#K2">-infty</a>  &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-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#K4" target="_self">-</a> <font color="Maroon">t</font>, <font color="Maroon">b</font> = <font color="Maroon">y</font> <a href="supinf_2.html#K4" target="_self">-</a> <font color="Maroon">s</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E3:18"/>
( ( <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="E4:18"/><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="E5:18"/><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="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_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="E2:18_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>, <a class="ref" href="supinf_2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:18_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="E4:18_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="E6:18_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="E7:18_1"/><i><font color="Green">E51</font></i>: 
 <a href="real_1.html#K1">-</a> <font color="Maroon">Ot</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">Os</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T1" target="_self">Th1</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E8:18_1"/>
( <font color="Maroon">Ox</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">Ot</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ox</font> <a href="real_1.html#K5">-</a> <font color="Maroon">Ot</font> &amp; <font color="Maroon">Oy</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">Os</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Oy</font> <a href="real_1.html#K5">-</a> <font color="Maroon">Os</font> )
 
;<br/>
<a NAME="E9:18_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">Ox</font> <a href="real_1.html#K5">-</a> <font color="Maroon">Ot</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">Oy</font> <a href="real_1.html#K5">-</a> <font color="Maroon">Os</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T1" target="_self">Th1</a>, <a class="txt" href="supinf_2.html#E1:18"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E10:18_1"/>
( <font color="Maroon">x</font> <a href="supinf_2.html#K4" target="_self">-</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ox</font> <a href="real_1.html#K5">-</a> <font color="Maroon">Ot</font> &amp; <font color="Maroon">y</font> <a href="supinf_2.html#K4" target="_self">-</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Oy</font> <a href="real_1.html#K5">-</a> <font color="Maroon">Os</font> )
 
<b>by </b><i><a class="ref" href="supinf_2.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E11:18_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="E6:18"/><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="18_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2"/><i><font color="Green">E54</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>  )
 ;<br/>

<a NAME="E2:18_2"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:18_2"/>
( <font color="Maroon">y</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#K1">+infty</a>  )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:18_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_2.html#T1" target="_self">Th1</a>, <a class="ref" href="supinf_1.html#T1">SUPINF_1:1</a>, <a class="ref" href="supinf_1.html#T6">SUPINF_1:6</a>, <a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:18"/><i><font color="Green">E56</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="18_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_3"/><i><font color="Green">E57</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>  )
 ;<br/>

<a NAME="E2:18_3"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:18_3"/>
( <font color="Maroon">x</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#K2">-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="E4:18_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="txt" href="supinf_2.html#E1:18"><i><font color="Green">E24</font></i></a>, <a class="ref" href="supinf_1.html#T1">SUPINF_1:1</a>, <a class="ref" href="supinf_1.html#T6">SUPINF_1:6</a>, <a class="ref" href="supinf_1.html#T23">SUPINF_1:23</a>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:18"/>
(  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="18_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_4"/><i><font color="Green">E59</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="E2:18_4"/><b>then </b><i><font color="Green">E60</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">s</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E3:18_4"/>
( <font color="Maroon">x</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#K2">-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="E4:18_4"/>
contradiction
 <b>by </b><i>, <a class="txt" href="supinf_2.html#E1:18"><i><font color="Green">E24</font></i></a>, <a class="ref" href="supinf_1.html#T23">SUPINF_1:23</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="E9:18"/>
<font color="Maroon">x</font> <a href="supinf_2.html#K4" target="_self">-</a> <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K4" target="_self">-</a> <font color="Maroon">s</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>
