<?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>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">r2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r1</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font> or <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r1</font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>and </b><br/><a NAME="E2:1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r2</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">F</font> or <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r2</font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/>

<a NAME="E3:1"/><i><font color="Green">E49</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">F2</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<a NAME="E1:1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r1</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r2</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E2:1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E3:1_1"/>
<font color="Maroon">F1</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K34">diffreal</a>  <a href="funcop_1.html#K3">.:</a> <font color="Maroon">F1</font>,<font color="Maroon">F2</font>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#D6">RVSUM_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F2</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">F2</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">F2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F1</font>
 ;<br/>

<a NAME="E2:1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">F2</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F2</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:1_2"/><b>then </b>
( <font color="Maroon">F1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">F2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">F2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T47">RVSUM_1:47</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:1"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">F2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">F1</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">F2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="integra1.html#T24">INTEGRA1:24</a></i>;<br/>
<a NAME="E6:1"/>
(  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">F</font></span>)</span> &amp;  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">F2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">r2</font> )
 
<b>by </b><i>, , <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a>, <a class="ref" href="rvsum_1.html#T106">RVSUM_1:106</a></i>;<br/>
<b>hence </b><a NAME="E7:1"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">F2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">r2</font>
 <b>by </b><i/>;<br/>


</div>
