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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a><br/> for <font color="Olive">a</font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Olive">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="supinf_1.html#K3">REAL</a> <br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>   st  <a href="finsop_1.html#K5">dom</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">p</font> holds <br/> ex <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>  st <br/>(  <a href="finsop_1.html#K5">dom</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">m</font> &amp; <font color="Olive">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">r</font> &amp; ( for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">r</font> holds <br/><font color="Olive">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> ) ) ) &amp;  <a href="finsop_1.html#K5">dom</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Olive">m</font> &amp; ( for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">q</font> holds <br/> ex <font color="Olive">s</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>  st <br/>(  <a href="finsop_1.html#K5">dom</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">q</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">s</font> holds <br/><font color="Olive">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> ) ) ) holds <br/> <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">q</font>;<br/>
<a NAME="E1:1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 0</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="supinf_1.html#K3">REAL</a> ;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> , <font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1"/><i><font color="Green">E36</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 0
 <b>and </b><br/><a NAME="E2:1_1_1"/>
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font> holds <br/> ex <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>  st <br/>(  <a href="finsop_1.html#K5">dom</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">r</font> &amp; ( for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">r</font> holds <br/><font color="Olive">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> ) )
 <b>and </b><br/><a NAME="E3:1_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E4:1_1_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">q</font> holds <br/> ex <font color="Olive">s</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>  st <br/>(  <a href="finsop_1.html#K5">dom</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 0 &amp; <font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">s</font> holds <br/><font color="Olive">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> ) )
 ;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</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>thus </b><a NAME="E6:1_1_1"/>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:1_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a>, <a class="ref" href="rvsum_1.html#T102">RVSUM_1:102</a></i>;<br/>
<a NAME="E3:1_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">q</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_1"><b>now </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="E1:1_1_1_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E2:1_1_1_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> ) </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:1_1_1_1_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 ;<br/><b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">z</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">s</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E7:1_1_1_1_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 0
 <b>and </b><br/><a NAME="E8:1_1_1_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E9:1_1_1_1_1_1"/>
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">s</font> holds <br/><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E10:1_1_1_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E11:1_1_1_1_1_1"/>
<font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a>, <a class="ref" href="rvsum_1.html#T102">RVSUM_1:102</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:1_1_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">q</font></span>)</span> <a href="funcop_1.html#K2">--&gt;</a> 0
 <b>by </b><i><a class="ref" href="funcop_1.html#T17">FUNCOP_1:17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:1_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="finseq_2.html#K2">|-&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="finseq_2.html#D2">FINSEQ_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:1_1_1_1"/>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T111">RVSUM_1:111</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:1_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<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">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="supinf_1.html#K3">REAL</a> ;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> , <font color="Maroon">q</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_2_1"/><i><font color="Green">E49</font></i>: 
(  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font> holds <br/> ex <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>  st <br/>(  <a href="finsop_1.html#K5">dom</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">r</font> &amp; ( for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">r</font> holds <br/><font color="Olive">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> ) ) ) )
 <b>and </b><br/><a NAME="E2:1_2_1"/><i><font color="Green">E50</font></i>: 
(  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; ( for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">q</font> holds <br/> ex <font color="Olive">s</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>  st <br/>(  <a href="finsop_1.html#K5">dom</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> &amp; <font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">s</font> holds <br/><font color="Olive">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> ) ) ) )
 ;<br/><b>set </b><font color="Maroon">a0</font> = <font color="Maroon">a</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><a NAME="E3:1_2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E4:1_2_1"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/><a NAME="E5:1_2_1"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/><b>then reconsider </b><font color="Maroon">a0</font> = <font color="Maroon">a</font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="supinf_1.html#K3">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/><b>reconsider </b><font color="Maroon">p0</font> = <font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/><a NAME="E8:1_2_1"/><i><font color="Green">E53</font></i>: 
(  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p0</font> holds <br/> ex <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>  st <br/>(  <a href="finsop_1.html#K5">dom</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; <font color="Maroon">p0</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">r</font> &amp; ( for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">r</font> holds <br/><font color="Olive">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a0</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:1_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E2:1_2_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E3:1_2_1_1"/><i><font color="Green">E54</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p0</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_2_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p0</font>
 ;<br/><a NAME="E2:1_2_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_2.html#T9">FINSEQ_2:9</a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E4:1_2_1_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E5:1_2_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E6:1_2_1_1_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">r</font> holds <br/><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">r</font> = <font color="Maroon">r</font>;<br/><b>thus </b><a NAME="E7:1_2_1_1_1"/>
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E8:1_2_1_1_1"/>
<font color="Maroon">p0</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>thus </b><a NAME="E9:1_2_1_1_1"/>
for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">r</font> holds <br/><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a0</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:1_2_1_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:1_2_1_1_1_1"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, , , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:1_2_1_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:1_2_1_1_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E5:1_2_1_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></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/>
<b>then </b><i><font color="Green">E62</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">a0</font> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T124">ZFMISC_1:124</a></i>
;<br/>
<b>thus </b><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">a0</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:1_2_1_1"/>
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p0</font> holds <br/> ex <font color="Olive">r</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>  st <br/>(  <a href="finsop_1.html#K5">dom</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; <font color="Maroon">p0</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">r</font> &amp; ( for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">r</font> holds <br/><font color="Olive">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a0</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> ) )
 ;<br/>


</div><b>end;</b></div><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>;<br/><b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</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>consider </b><font color="Maroon">an</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E11:1_2_1"/><i><font color="Green">E65</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">an</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">an</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i><a NAME="E12:1_2_1"/>
<font color="Maroon">an</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</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_1_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">an</font>
 ;<br/><a NAME="E2:1_2_1_2_1"/><b>then </b>
<font color="Maroon">i</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#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:1_2_1_2_1"/><b>then </b>
<font color="Maroon">an</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:1_2_1_2_1"/>
<font color="Maroon">an</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:1_2_1_2"/>
<font color="Maroon">an</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i><a class="ref" href="finseq_2.html#T14">FINSEQ_2:14</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">an</font> = <font color="Maroon">an</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>  ;<br/><a NAME="E14:1_2_1"/><i><font color="Green">E66</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">an</font> <a href="hidden.html#R1">=</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#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E15:1_2_1"/><i><font color="Green">E67</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">an</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:1_2_1_3"/>
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E3:1_2_1_3"/><i><font color="Green">E68</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E4:1_2_1_3"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E5:1_2_1_3"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">r</font> holds <br/><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</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_1_3_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">r</font>
 ;<br/><b>hence </b><font color="Maroon">r</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">an</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:1_2_1_3"/>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">an</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">q0</font> = <font color="Maroon">q</font> <a href="rvsum_1.html#K7">-</a> <font color="Maroon">an</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>  ;<br/><a NAME="E17:1_2_1"/><i><font color="Green">E73</font></i>: 
(  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">q0</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; ( for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">q0</font> holds <br/> ex <font color="Olive">s</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>  st <br/>(  <a href="finsop_1.html#K5">dom</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; <font color="Maroon">q0</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">s</font> holds <br/><font color="Olive">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a0</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:1_2_1_4"/><i><font color="Green">E74</font></i>: 
 <a href="relset_1.html#K4">dom</a> <a href="binop_2.html#K34">diffreal</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="supinf_1.html#K3">REAL</a> ,<a href="supinf_1.html#K3">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="E2:1_2_1_4"/><i><font color="Green">E75</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">q</font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">an</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#K34">diffreal</a> 
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E3:1_2_1_4"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">an</font></span><a href="funct_3.html#K13">:&gt;</a></span> <a href="tarski.html#R1">c=</a> <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">q</font></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">an</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#T71">FUNCT_3:71</a></i>;<br/>
<a NAME="E4:1_2_1_4"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">an</font></span><a href="funct_3.html#K13">:&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <a href="binop_2.html#K34">diffreal</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<i><font color="Green">E77</font></i>:  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K34">diffreal</a>  <a href="finseqop.html#K1">.:</a> <font color="Maroon">q</font>,<font color="Maroon">an</font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K34">diffreal</a>  <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">an</font></span><a href="funct_3.html#K13">:&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#D3">FUNCOP_1:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_3.html#K13">&lt;:</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">an</font></span><a href="funct_3.html#K13">:&gt;</a></span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">q</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">an</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_3.html#D8">FUNCT_3:def 8</a></i>
;<br/>
<b>thus </b><i><font color="Green">E78</font></i>:  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">q0</font> = 
<span class="p1">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">q</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="finsop_1.html#K5">dom</a> <font color="Maroon">an</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rvsum_1.html#D6">RVSUM_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</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/>.= 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>

;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1_4_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_2_1_4_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">q0</font>
 ;<br/><b>consider </b><font color="Maroon">s</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a> <b> such that </b><br/><a NAME="E3:1_2_1_4_3"/><i><font color="Green">E80</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E4:1_2_1_4_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E5:1_2_1_4_3"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">s</font> holds <br/><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, , </i>;<br/><b>reconsider </b><font color="Maroon">sn</font> = <font color="Maroon">s</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/><b>take </b><font color="Maroon">sn</font> = <font color="Maroon">sn</font>;<br/><a NAME="E7:1_2_1_4_3"/><i><font color="Green">E84</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:1_2_1_4_3"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><a NAME="E9:1_2_1_4_3"/><i><font color="Green">E85</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">sn</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/><a NAME="E10:1_2_1_4_3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">sn</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>;<br/><a NAME="E11:1_2_1_4_3"/>
<font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/><a NAME="E12:1_2_1_4_3"/><b>then </b>
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E13:1_2_1_4_3"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">an</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , </i>;<br/><i><font color="Green">E88</font></i>: <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">an</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">sn</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">an</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">an</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="mesfunc3.html#T1" target="_self">Th1</a>, <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">sn</font>

;<br/><a NAME="E15:1_2_1_4_3"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <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>hence </b><a NAME="E16:1_2_1_4_3"/>
<font color="Maroon">q0</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">sn</font>
 <b>by </b><i>, , <a class="ref" href="rvsum_1.html#T47">RVSUM_1:47</a></i>;<br/><b>thus </b><a NAME="E17:1_2_1_4_3"/>
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">sn</font> holds <br/><font color="Maroon">sn</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a0</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_1_4_3_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:1_2_1_4_3_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">sn</font>
 ;<br/>

<a NAME="E2:1_2_1_4_3_2"/><i><font color="Green">E90</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, , , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:1_2_1_4_3_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E4:1_2_1_4_3_2"/><b>then </b><i><font color="Green">E91</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E5:1_2_1_4_3_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></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/>
<b>then </b><i><font color="Green">E92</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">a0</font> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T124">ZFMISC_1:124</a></i>
;<br/>
<b>thus </b><font color="Maroon">sn</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">s</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , , , </i>
<br/>.= 
<font color="Maroon">a0</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:1_2_1_4"/>
for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">q0</font> holds <br/> ex <font color="Olive">s</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="supinf_1.html#K3">REAL</a>  st <br/>(  <a href="finsop_1.html#K5">dom</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; <font color="Maroon">q0</font> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">s</font> holds <br/><font color="Olive">s</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a0</font> <a href="seq_1.html#K2">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> ) )
 ;<br/>


</div><b>end;</b></div><a NAME="E18:1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E19:1_2_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">p0</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">an</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>;<br/><a NAME="E20:1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E21:1_2_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">q0</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/><a NAME="E22:1_2_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">an</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/><a NAME="E23:1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E24:1_2_1"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">q</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">m</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="supinf_1.html#K3">REAL</a> 
 <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/><a NAME="E25:1_2_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">q0</font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">an</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 <b>by </b><i>, , <a class="ref" href="rvsum_1.html#T64">RVSUM_1:64</a></i>;<br/><b>thus </b> <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">p0</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">an</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">q0</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">an</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">q</font>
<b>by </b><i>, , , <a class="ref" href="rvsum_1.html#T119">RVSUM_1:119</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:1"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );</i><br/>


</div>
