<?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> ;<br/><b>let </b><font color="Maroon">G</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">H</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:45"/><i><font color="Green">E49</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> )
 <b>and </b><br/><a NAME="E2:45"/><i><font color="Green">E53</font></i>: 
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">F</font> holds <br/><font color="Maroon">H</font> <a href="goboard1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font></span>)</span>
 ;<br/>

<a NAME="E3:45"/><i><font color="Green">E54</font></i>: 
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">F</font> holds <br/><font color="Maroon">H</font> <a href="goboard1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="goboard1.html#K1">.</a> <font color="Olive">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><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="E1:45_1"/><i><font color="Green">E55</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">F</font>
 ;<br/>

<a NAME="E2:45_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">F</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="goboard1.html#K1">.</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/>
<a NAME="E3:45_1"/>
<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">G</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:45_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:45_1"/><b>then </b>
<font color="Maroon">G</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="goboard1.html#K1">.</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="E6:45_1"/>
<font color="Maroon">H</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:45"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">F</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E5:45"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">G</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E6:45"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">F</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">G</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i>, , <a class="ref" href="finseq_2.html#T140">FINSEQ_2:140</a></i>;<br/>
<a NAME="E7:45"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E8:45"/>
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#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">G</font></span>)</span></span>)</span> holds <br/><font color="Maroon">H</font> <a href="goboard1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">G</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_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="E1:45_2"/><i><font color="Green">E65</font></i>: 
<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> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:45_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">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E3:45_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:45_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">H</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:45_2"/>
<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">F</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:45_2"/>
<font color="Maroon">H</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">G</font></span>)</span> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T47">RVSUM_1:47</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b> <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">H</font> = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">F</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="rvsum_1.html#T120">RVSUM_1:120</a></i>
;<br/>
<b>hence </b><a NAME="E10:45"/>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">H</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="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">G</font></span>)</span>
 ;<br/>


</div>
