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

<b>let </b><font color="Maroon">a</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">s</font> be   <a href="afinsq_1.html#NM2">XFinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green">E25</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> &amp; <font color="Maroon">s</font> <a href="prgcor_2.html#K6" target="_self">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> holds <br/><font color="Maroon">s</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Olive">i</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> implies  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E2:17"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T102">RVSUM_1:102</a></i>;<br/>
<a NAME="E3:17"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">k1</font> <a href="nat_1.html#K1">+</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E4:17_1"/>
( <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font> implies  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:17_1_1"/><i><font color="Green">E43</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:17_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:17_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="jordan8.html#T2">JORDAN8:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q0</font> = <font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">rx</font> = <font color="Maroon">a</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">x0</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E8:17_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:17_1_1"/><b>then </b>
<font color="Maroon">m</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">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E10:17_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">m</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">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green">E49</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q0</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">rx</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q0</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<a NAME="E12:17_1_1"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E13:17_1_1"/>
<font color="Maroon">m</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">a</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:17_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green">E61</font></i>: <font color="Maroon">rx</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
<b>by </b><i>, , <a class="ref" href="prgcor_2.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:17"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<a NAME="E16:17_1_1"/><i><font color="Green">E62</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">rx</font>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T103">RVSUM_1:103</a></i>;<br/>
<a NAME="E17:17_1_1"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="prgcor_2.html#K6" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">rx</font>
 
<b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a>, , , <a class="ref" href="rvsum_1.html#T105">RVSUM_1:105</a></i>;<br/>
<b>hence </b><a NAME="E18:17_1_1"/>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="txt" href="prgcor_2.html#E2:17"><i><font color="Green">E26</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:17_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:17"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<a NAME="E5:17"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font>]
 
;<br/>
<b>hence </b><a NAME="E6:17"/>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="prgcor_2.html#K6" target="_self">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/>


</div>
