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

<b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">F1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">F2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:10_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="finseqop.html#K1">.:</a> <font color="Maroon">F1</font>,<font color="Maroon">F2</font>
 ;<br/>

<a NAME="E2:10_1_1"/><i><font color="Green">E51</font></i>: 
 <a href="relset_1.html#K4">dom</a> <a href="binop_2.html#K33">addreal</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:10_1_1"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F2</font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F1</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="binop_2.html#K33">addreal</a> 
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E4:10_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F1</font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F2</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="binop_2.html#K33">addreal</a> 
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>then </b><i><font color="Green">E61</font></i>:  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K33">addreal</a>  <a href="finseqop.html#K1">.:</a> <font color="Maroon">F1</font>,<font color="Maroon">F2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">F1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K3">dom</a> <font color="Maroon">F2</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T84">FUNCOP_1:84</a></i>
<br/>.= 
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K33">addreal</a>  <a href="finseqop.html#K1">.:</a> <font color="Maroon">F2</font>,<font color="Maroon">F1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T84">FUNCOP_1:84</a></i>
;<br/>
<a NAME="E6:10_1_1"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K33">addreal</a>  <a href="finseqop.html#K1">.:</a> <font color="Maroon">F2</font>,<font color="Maroon">F1</font></span>)</span> holds <br/><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">z</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seq_1.html#K2">.</a> <font color="Olive">z</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:10_1_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K33">addreal</a>  <a href="finseqop.html#K1">.:</a> <font color="Maroon">F2</font>,<font color="Maroon">F1</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">F1z</font> = <font color="Maroon">F1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font>;<br/>
<b>set </b><font color="Maroon">F2z</font> = <font color="Maroon">F2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font>;<br/>
<b>thus </b><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font> = 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funcop_1.html#T28">FUNCOP_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="binop_2.html#D9">BINOP_2:def 9</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:10_1_1"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="finseqop.html#K1">.:</a> <font color="Maroon">F2</font>,<font color="Maroon">F1</font>
 <b>by </b><i>, , <a class="ref" href="funcop_1.html#T27">FUNCOP_1:27</a></i>;<br/>


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