<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> , <font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<a NAME="E1:80"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K4">-</a> <font color="Maroon">X</font></span>)</span> <a href="complsp1.html#K21">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4">-</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="real_1.html#K3">+</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">y</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K4">-</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K4">-</a> <font color="Maroon">Y</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="complsp1.html#D21">COMPLSP1:def 21</a></i>;<br/>
<a NAME="E2:80"/><i><font color="Green">E53</font></i>: 
 <a href="pscomp_1.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="complsp1.html#K21">+</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Olive">z</font></span>)</span> where <font color="Olive">z</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="complsp1.html#K21">+</a> <font color="Maroon">Y</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D7">PSCOMP_1:def 7</a></i>;<br/>
<a NAME="E3:80"/><i><font color="Green">E54</font></i>: 
 <a href="pscomp_1.html#K4">-</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Olive">z</font></span>)</span> where <font color="Olive">z</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D7">PSCOMP_1:def 7</a></i>;<br/>
<a NAME="E4:80"/><i><font color="Green">E55</font></i>: 
 <a href="pscomp_1.html#K4">-</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Olive">z</font></span>)</span> where <font color="Olive">z</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D7">PSCOMP_1:def 7</a></i>;<br/>
<a NAME="E5:80"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">X</font> <a href="complsp1.html#K21">+</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="real_1.html#K3">+</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">y</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="complsp1.html#D21">COMPLSP1:def 21</a></i>;<br/>
<a NAME="E6:80"/>
for <font color="Olive">z</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4">-</a> <font color="Maroon">X</font></span>)</span> <a href="complsp1.html#K21">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4">-</a> <font color="Maroon">Y</font></span>)</span> holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="complsp1.html#K21">+</a> <font color="Maroon">Y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:80_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4">-</a> <font color="Maroon">X</font></span>)</span> <a href="complsp1.html#K21">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4">-</a> <font color="Maroon">Y</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">y</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:80_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K4">-</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K4">-</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="txt" href="integra1.html#E8:80_1"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:80_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:80_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:80_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <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>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E9:80_1"/>
<font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="complsp1.html#K21">+</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E7:80"><i><font color="Green">E65</font></i></a>, , </i>;<br/>
<b>hence </b><a NAME="E10:80_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="complsp1.html#K21">+</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="integra1.html#T6" target="_self">Th6</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:80"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K4">-</a> <font color="Maroon">X</font></span>)</span> <a href="complsp1.html#K21">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4">-</a> <font color="Maroon">Y</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pscomp_1.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="complsp1.html#K21">+</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>
<a NAME="E8:80"/>
for <font color="Olive">z</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="complsp1.html#K21">+</a> <font color="Maroon">Y</font></span>)</span> holds <br/><font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4">-</a> <font color="Maroon">X</font></span>)</span> <a href="complsp1.html#K21">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4">-</a> <font color="Maroon">Y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:80_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="complsp1.html#K21">+</a> <font color="Maroon">Y</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:80_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="complsp1.html#K21">+</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="integra1.html#T6" target="_self">Th6</a></i>;<br/>
<b>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><b> such that </b><br/><a NAME="E5:80_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="txt" href="integra1.html#E7:80"><i><font color="Green">E65</font></i></a>, <a class="txt" href="integra1.html#E5:80_2"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E6:80_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E5:80_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="integra1.html#E6:80_2"><i><font color="Green">E68</font></i></a></i>;<br/>
<a NAME="E7:80_2"/>
(  <a href="real_1.html#K1">-</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K4">-</a> <font color="Maroon">X</font> &amp;  <a href="real_1.html#K1">-</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K4">-</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, , <a class="txt" href="integra1.html#E6:80_2"><i><font color="Green">E68</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:80_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4">-</a> <font color="Maroon">X</font></span>)</span> <a href="complsp1.html#K21">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4">-</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:80"/><b>then </b>
 <a href="pscomp_1.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="complsp1.html#K21">+</a> <font color="Maroon">Y</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4">-</a> <font color="Maroon">X</font></span>)</span> <a href="complsp1.html#K21">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4">-</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>
<b>hence </b><a NAME="E10:80"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K4">-</a> <font color="Maroon">X</font></span>)</span> <a href="complsp1.html#K21">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4">-</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K4">-</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="complsp1.html#K21">+</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
