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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<a NAME="E1:79"/><i><font color="Green" title="E60">A1</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="complsp1.html#D21" title="COMPLSP1:def.21">COMPLSP1:def 21</a></i>;<br/>
<a NAME="E2:79"/><i><font color="Green" title="E61">A2</font></i>: 
 <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Olive" title="b1">z</font></span>)</span> where <font color="Olive" title="b1">z</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D7" title="PSCOMP_1:def.7">PSCOMP_1:def 7</a></i>;<br/>
<a NAME="E3:79"/><i><font color="Green" title="E62">A3</font></i>: 
 <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Olive" title="b1">z</font></span>)</span> where <font color="Olive" title="b1">z</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D7" title="PSCOMP_1:def.7">PSCOMP_1:def 7</a></i>;<br/>
<a NAME="E4:79"/><i><font color="Green" title="E63">A4</font></i>: 
 <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Olive" title="b1">z</font></span>)</span> where <font color="Olive" title="b1">z</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#D7" title="PSCOMP_1:def.7">PSCOMP_1:def 7</a></i>;<br/>
<a NAME="E5:79"/><i><font color="Green" title="E64">A5</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="complsp1.html#D21" title="COMPLSP1:def.21">COMPLSP1:def 21</a></i>;<br/>
<a NAME="E6:79"/>
for <font color="Olive" title="b1">z</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span> holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">z</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:79_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:79_1"/><i><font color="Green" title="E65">A6</font></i>: 
( <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font> )
 <b>by </b><i><a class="txt" href="integra1.html#E1:79"><i><font color="Green" title="E60">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:79_1"/><i><font color="Green" title="E66">A7</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="txt" href="integra1.html#E3:79"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="integra1.html#E3:79_1"><i><font color="Green" title="E65">A6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:79_1"/><i><font color="Green" title="E67">A8</font></i>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c7">b</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 <b>by </b><i><a class="txt" href="integra1.html#E4:79"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="integra1.html#E3:79_1"><i><font color="Green" title="E65">A6</font></i></a></i>;<br/>
<a NAME="E8:79_1"/><i><font color="Green" title="E68">A9</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c7">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:79_1"><i><font color="Green" title="E65">A6</font></i></a>, <a class="txt" href="integra1.html#E5:79_1"><i><font color="Green" title="E66">A7</font></i></a>, <a class="txt" href="integra1.html#E7:79_1"><i><font color="Green" title="E67">A8</font></i></a></i>;<br/>
<a NAME="E9:79_1"/>
<font color="Maroon" title="c6">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="txt" href="integra1.html#E5:79"><i><font color="Green" title="E64">A5</font></i></a>, <a class="txt" href="integra1.html#E5:79_1"><i><font color="Green" title="E66">A7</font></i></a>, <a class="txt" href="integra1.html#E7:79_1"><i><font color="Green" title="E67">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:79_1"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E2:79"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="integra1.html#E8:79_1"><i><font color="Green" title="E68">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:79"/><b>then </b><i><font color="Green" title="E65">A10</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>
<a NAME="E8:79"/>
for <font color="Olive" title="b1">z</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font></span>)</span> holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">z</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:79_2"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">x</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:79_2"/><i><font color="Green" title="E66">A11</font></i>: 
( <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font> )
 <b>by </b><i><a class="txt" href="integra1.html#E2:79"><i><font color="Green" title="E61">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:79_2"/><i><font color="Green" title="E67">A12</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> )
 <b>by </b><i><a class="txt" href="integra1.html#E5:79"><i><font color="Green" title="E64">A5</font></i></a>, <a class="txt" href="integra1.html#E3:79_2"><i><font color="Green" title="E66">A11</font></i></a></i>;<br/>
<a NAME="E6:79_2"/><i><font color="Green" title="E68">A13</font></i>: 
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c6">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="integra1.html#E3:79_2"><i><font color="Green" title="E66">A11</font></i></a>, <a class="txt" href="integra1.html#E5:79_2"><i><font color="Green" title="E67">A12</font></i></a></i>;<br/>
<a NAME="E7:79_2"/>
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font> )
 
<b>by </b><i><a class="txt" href="integra1.html#E3:79"><i><font color="Green" title="E62">A3</font></i></a>, <a class="txt" href="integra1.html#E4:79"><i><font color="Green" title="E63">A4</font></i></a>, <a class="txt" href="integra1.html#E5:79_2"><i><font color="Green" title="E67">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:79_2"/>
<font color="Maroon" title="c3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E1:79"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="integra1.html#E6:79_2"><i><font color="Green" title="E68">A13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:79"/><b>then </b>
 <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>
<b>hence </b><a NAME="E10:79"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K4" title="PSCOMP_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="complsp1.html#K21" title="COMPLSP1:func.21">+</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <b>by </b><i><a class="txt" href="integra1.html#E7:79"><i><font color="Green" title="E65">A10</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
