<?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>;<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><a NAME="E1:1"/><i><font color="Green">E23</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp;  <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> )
 ;<br/>

<a NAME="E2:1"/><i><font color="Green">E24</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> ) &amp; ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">z</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="xxreal_0.html#D4">XXREAL_0:def 4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1"/>
( ( <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">z</font> is   <a href="real_1.html#NM1">Real</a> ) or <font color="Maroon">x</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#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span><b>by </b><i>, , <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1"/><i><font color="Green">E25</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">z</font> is   <a href="real_1.html#NM1">Real</a> )
 ;<br/><div class="add"><a NAME="E2:1_1_1"/>
( <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> is   <a href="real_1.html#NM1">Real</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:1_1_1_1"/>
( 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">x</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+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">y</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; not <font color="Maroon">z</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#K1">+infty</a>  )
 
<b>by </b><i>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>
<a NAME="E2:1_1_1_1"/><b>then </b>
( not <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>  &amp; not <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#K1">+infty</a>  &amp; not <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; not <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</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#T8">SUPINF_2:8</a>, <a class="ref" href="supinf_2.html#T9">SUPINF_2:9</a></i>;<br/>
<b>hence </b><a NAME="E3:1_1_1_1"/>
( <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> is   <a href="real_1.html#NM1">Real</a> )
 <b>by </b><i><a class="ref" href="measure3.html#T2">MEASURE3:2</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">c</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">d</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">e</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <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">e</font> )
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> = 
<font color="Maroon">d</font> <a href="real_1.html#K3">+</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>
<br/>.= 
<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#K3">+</a> <font color="Maroon">c</font>
<b>by </b><i>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <font color="Maroon">c</font></span>)</span>

<br/>.= 
<font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <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>
<b>by </b><i>, <a class="ref" href="supinf_2.html#T1">SUPINF_2:1</a></i>
;<br/><b>hence </b><a NAME="E6:1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <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>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_2"><b>suppose </b></a><a NAME="E1:1_1_2"/><i><font color="Green">E32</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="E2:1_1_2"/><i><font color="Green">E33</font></i>: 
not <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <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="supinf_2.html#T9">SUPINF_2:9</a>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> = 
<a href="supinf_1.html#K1">+infty</a>  <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font>
<b>by </b><i>, , <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>
<br/>.= 
 <a href="supinf_1.html#K1">+infty</a> 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>
<br/>.= 
<a href="supinf_1.html#K1">+infty</a>  <a href="supinf_2.html#K2">+</a> <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>
<b>by </b><i>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>
;<br/><b>hence </b><a NAME="E4:1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <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>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_3"><b>suppose </b></a><a NAME="E1:1_1_3"/><i><font color="Green">E34</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"><b>then </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> = 
<a href="supinf_1.html#K1">+infty</a>  <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font>
<b>by </b><i>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>
<br/>.= 
 <a href="supinf_1.html#K1">+infty</a> 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <a href="supinf_1.html#K1">+infty</a> 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K1">+infty</a>  <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>
;<br/><b>hence </b><a NAME="E3:1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <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>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_4"><b>suppose </b></a><a NAME="E1:1_1_4"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:1_1_4"/>
not <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="ref" href="supinf_2.html#T9">SUPINF_2:9</a>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> = 
 <a href="supinf_1.html#K1">+infty</a> 
<b>by </b><i>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <a href="supinf_1.html#K1">+infty</a> 
<b>by </b><i>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <a href="supinf_1.html#K1">+infty</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>
;<br/><b>hence </b><a NAME="E4:1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <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>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
