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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K1">REAL</a> , <font color="Maroon">g</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">X</font>, <a href="numbers.html#K1">REAL</a> ;<br/>



<a NAME="E1:33"/>
for <font color="Olive">y</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> holds <br/><font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="complsp1.html#K21">+</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:33_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:33_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/>
<a NAME="E5:33_1"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:33_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:33_1"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="complsp1.html#K21">+</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</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">z</font></span>)</span> where <font color="Olive">x</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">z</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="complsp1.html#D21">COMPLSP1:def 21</a></i>;<br/>
<a NAME="E8:33_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:33_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="complsp1.html#K21">+</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:33"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="complsp1.html#K21">+</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>


</div>
