<?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">z</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:3"/><i><font color="Green">E21</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 ;<br/>

<a NAME="E2:3"/><b>then </b>
<a href="supinf_2.html#K1">0.</a>  <a href="supinf_2.html#K2">+</a> <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:3"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/>
<a NAME="E4:3"/>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<a NAME="E1:3_1"/><i><font color="Green">E25</font></i>: 
( not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; not <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; not <font color="Maroon">z</font> <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#T12">XXREAL_0:12</a></i>;<br/>
<a NAME="E2:3_1"/><i><font color="Green">E26</font></i>: 
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">x</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">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">y</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>, <a class="ref" href="xxreal_0.html#D4">XXREAL_0:def 4</a></i>;<br/>
<a NAME="E3:3_1"/><i><font color="Green">E27</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">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  implies <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">y</font>, <font color="Maroon">b</font> = <font color="Maroon">z</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, , <a class="ref" href="supinf_2.html#T12">SUPINF_2:12</a></i>;<br/>
<a NAME="E3:3_1_1"/>
<font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="supinf_2.html#T5">SUPINF_2:5</a></i>
<br/>.= 
<font color="Maroon">z</font>

;<br/>
<b>hence </b><a NAME="E5:3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:3_1"/>
( <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#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  implies <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_2"/><i><font color="Green">E30</font></i>: 
( <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#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font> = 
 <a href="supinf_1.html#K1">+infty</a> 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T6">SUPINF_2:6</a></i>
<br/>.= 
<font color="Maroon">z</font>
<b>by </b><i>, , <a class="ref" href="supinf_2.html#T8">SUPINF_2:8</a></i>
;<br/>
<b>hence </b><a NAME="E3:3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:3"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>


</div>
