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

<b>let </b><font color="Maroon" title="c1">b</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c3">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">m</font> )
 ;<br/>

<a NAME="E2:19"/><b>then </b><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/>
<a NAME="E3:19"/><i><font color="Green" title="E16">A3</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">b</font>
 
<b>by </b><i><a class="ref" href="prgcor_2.html#T2" target="_self" title="PRGCOR_2:th.2">Th2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">k</font> = <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="prgcor_2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">c2</font> = <font color="Maroon" title="c2">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E6:19"/><i><font color="Green" title="E17">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="prgcor_2.html#D4" target="_self" title="PRGCOR_2:def.4">Def4</a></i>;<br/>
<a NAME="E7:19"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>;<br/>
<a NAME="E8:19"/><b>then </b>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p3">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:19"/><b>then </b><i><font color="Green" title="E18">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E6:19"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">p</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E11:19"/><i><font color="Green" title="E19">A6</font></i>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c6">p</font> <a href="prgcor_2.html#R1" title="PRGCOR_2:pred.1">is_an_xrep_of</a> <font color="Maroon" title="c5">c2</font> )
 <b>by </b><i><a class="txt" href="prgcor_2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="prgcor_2.html#T7" target="_self" title="PRGCOR_2:th.7">Th7</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">p2</font> =  <a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">Replace</a> <font color="Maroon" title="c6">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>;<br/>
<a NAME="E12:19"/><i><font color="Green" title="E20">A7</font></i>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">Replace</a> <font color="Maroon" title="c6">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T48" title="AFINSQ_1:th.48">AFINSQ_1:48</a></i>;<br/>
<a NAME="E13:19"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <span class="p1">(<span class="default"><a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">Replace</a> <font color="Maroon" title="c6">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">Replace</a> <font color="Maroon" title="c6">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<a NAME="E1:19_1"/><i><font color="Green" title="E21">A8</font></i>: 
( <font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> holds <br/><span class="p1">(<span class="default"><a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">Replace</a> <font color="Maroon" title="c6">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1_1_1"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>thus </b><a NAME="E2:19_1_1_1"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> holds <br/><span class="p1">(<span class="default"><a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">Replace</a> <font color="Maroon" title="c6">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<a NAME="E1:19_1_1_1_1"/><i><font color="Green" title="E22">A9</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:19_1_1_1_1"/><i><font color="Green" title="E23">A10</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> )
 ;<br/>

<a NAME="E3:19_1_1_1_1"/><b>then </b><i><font color="Green" title="E24">A11</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E9:19"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="prgcor_2.html#E11:19"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="prgcor_2.html#D5" target="_self" title="PRGCOR_2:def.5">Def5</a></i>;<br/>
<a NAME="E4:19_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="prgcor_2.html#E2:19_1_1_1_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="prgcor_2.html#D4" target="_self" title="PRGCOR_2:def.4">Def4</a></i>;<br/>
<a NAME="E5:19_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T66" title="RVSUM_1:th.66">RVSUM_1:66</a></i>;<br/>
<b>hence </b><a NAME="E6:19_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">Replace</a> <font color="Maroon" title="c6">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E1:19_1_1_1_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="prgcor_2.html#E2:19_1_1_1_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="prgcor_2.html#E3:19_1_1_1_1"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="afinsq_1.html#T48" title="AFINSQ_1:th.48">AFINSQ_1:48</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:19_1_1"/>
( <font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> holds <br/><span class="p1">(<span class="default"><a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">Replace</a> <font color="Maroon" title="c6">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:19_1"/>
<span class="p1">(<span class="default"><a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">Replace</a> <font color="Maroon" title="c6">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E11:19"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="afinsq_1.html#T48" title="AFINSQ_1:th.48">AFINSQ_1:48</a></i>;<br/>
<b>hence </b><a NAME="E3:19_1"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <span class="p1">(<span class="default"><a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">Replace</a> <font color="Maroon" title="c6">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">Replace</a> <font color="Maroon" title="c6">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="prgcor_2.html#E1:19_1"><i><font color="Green" title="E21">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:19"/><b>then </b>
<font color="Maroon" title="c3">m</font> <a href="prgcor_2.html#R2" title="PRGCOR_2:pred.2">scalar_prd_prg</a>  <a href="afinsq_1.html#K11" title="AFINSQ_1:func.11">Replace</a> <font color="Maroon" title="c6">p</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E11:19"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="prgcor_2.html#E12:19"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="prgcor_2.html#D8" target="_self" title="PRGCOR_2:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E15:19"/>
 ex <font color="Olive" title="b1">c</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Maroon" title="c3">m</font> <a href="prgcor_2.html#R2" title="PRGCOR_2:pred.2">scalar_prd_prg</a> <font color="Olive" title="b1">c</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>
 ;<br/>

<b>thus </b><a NAME="E16:19"/>
for <font color="Olive" title="b1">c</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c3">m</font> <a href="prgcor_2.html#R2" title="PRGCOR_2:pred.2">scalar_prd_prg</a> <font color="Olive" title="b1">c</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font> holds <br/> <a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">c</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:19_2"/><i><font color="Green" title="E21">A12</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="prgcor_2.html#R2" title="PRGCOR_2:pred.2">scalar_prd_prg</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c1">b</font>
 ;<br/>

<a NAME="E2:19_2"/><b>then </b><i><font color="Green" title="E22">A13</font></i>: 
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> &amp;  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> &amp;  ex <font color="Olive" title="b1">n</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Maroon" title="c8">c</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c8">c</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">i</font></span>)</span> ) ) )
 
<b>by </b><i><a class="ref" href="prgcor_2.html#D8" target="_self" title="PRGCOR_2:def.8">Def8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">n</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E4:19_2"/><i><font color="Green" title="E23">A14</font></i>: 
( <font color="Maroon" title="c8">c</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; ( <font color="Maroon" title="c9">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> holds <br/><font color="Maroon" title="c8">c</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="prgcor_2.html#E1:19_2"><i><font color="Green" title="E21">A12</font></i></a>, <a class="ref" href="prgcor_2.html#D8" target="_self" title="PRGCOR_2:def.8">Def8</a></i>;<br/>
<a NAME="E5:19_2"/><i><font color="Green" title="E24">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="prgcor_2.html#E2:19_2"><i><font color="Green" title="E22">A13</font></i></a>, <a class="ref" href="prgcor_2.html#D4" target="_self" title="PRGCOR_2:def.4">Def4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1_1"/>
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c9">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_1"><b>case </b></a><a NAME="E1:19_2_1_1_1"/><i><font color="Green" title="E25">A16</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:19_2_1_1_1"/><b>then </b><i><font color="Green" title="E26">A17</font></i>: 
 <a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="txt" href="prgcor_2.html#E3:19"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="prgcor_2.html#E4:19_2"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="prgcor_2.html#T5" target="_self" title="PRGCOR_2:th.5">Th5</a></i>;<br/><a NAME="E3:19_2_1_1_1"/>
 <a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="txt" href="prgcor_2.html#E3:19"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="prgcor_2.html#E2:19_2"><i><font color="Green" title="E22">A13</font></i></a>, <a class="txt" href="prgcor_2.html#E4:19_2"><i><font color="Green" title="E23">A14</font></i></a>, <a class="txt" href="prgcor_2.html#E1:19_2_1_1_1"><i><font color="Green" title="E25">A16</font></i></a>, <a class="ref" href="prgcor_2.html#T5" target="_self" title="PRGCOR_2:th.5">Th5</a></i>;<br/><b>hence </b><a NAME="E4:19_2_1_1_1"/>
 <a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E2:19_2_1_1_1"><i><font color="Green" title="E26">A17</font></i></a>, <a class="ref" href="rvsum_1.html#T68" title="RVSUM_1:th.68">RVSUM_1:68</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1_2"><b>case </b></a><a NAME="E1:19_2_1_1_2"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c10">p3</font> =  <a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c8">c</font>;<br/><a NAME="E2:19_2_1_1_2"/>
for <font color="Olive" title="b1">k3</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k3</font> &amp; <font color="Olive" title="b1">k3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c8">c</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">k3</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:19_2_1_1_2_1"/><i><font color="Green" title="E25">A18</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k3</font> &amp; <font color="Maroon" title="c11">k3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c8">c</font></span>)</span> )
 ;<br/>

<a NAME="E2:19_2_1_1_2_1"/><i><font color="Green" title="E26">A19</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c8">c</font>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E2:19_2"><i><font color="Green" title="E22">A13</font></i></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></i>;<br/>
<a NAME="E3:19_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E27">A20</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> holds <br/><span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">c</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E4:19_2"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="prgcor_2.html#D4" target="_self" title="PRGCOR_2:def.4">Def4</a></i>;<br/>
<a NAME="E4:19_2_1_1_2_1"/><i><font color="Green" title="E28">A21</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k3</font> &amp; <font color="Maroon" title="c11">k3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> )
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E4:19_2"><i><font color="Green" title="E23">A14</font></i></a>, <a class="txt" href="prgcor_2.html#E1:19_2_1_1_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="prgcor_2.html#E2:19_2_1_1_2_1"><i><font color="Green" title="E26">A19</font></i></a>, <a class="ref" href="prgcor_2.html#D4" target="_self" title="PRGCOR_2:def.4">Def4</a></i>;<br/>
<i><font color="Green" title="E29">A22</font></i>: <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k3</font> = 
<font color="Maroon" title="c8">c</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k3</font>
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:19_2_1_1_2_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="prgcor_2.html#E3:19_2_1_1_2_1"><i><font color="Green" title="E27">A20</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k3</font></span>)</span>
<b>by </b><i><a class="txt" href="prgcor_2.html#E4:19_2"><i><font color="Green" title="E23">A14</font></i></a>, <a class="txt" href="prgcor_2.html#E4:19_2_1_1_2_1"><i><font color="Green" title="E28">A21</font></i></a></i>
;<br/>
<a NAME="E6:19_2_1_1_2_1"/>
<font color="Maroon" title="c1">b</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k3</font>
 
<b>by </b><i><a class="txt" href="prgcor_2.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="prgcor_2.html#E2:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="prgcor_2.html#E4:19_2"><i><font color="Green" title="E23">A14</font></i></a>, <a class="txt" href="prgcor_2.html#E4:19_2_1_1_2_1"><i><font color="Green" title="E28">A21</font></i></a>, <a class="ref" href="prgcor_2.html#D4" target="_self" title="PRGCOR_2:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E7:19_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c8">c</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">k3</font>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E5:19_2_1_1_2_1"><i><font color="Green" title="E29">A22</font></i></a>, <a class="ref" href="rvsum_1.html#T66" title="RVSUM_1:th.66">RVSUM_1:66</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:19_2_1_1_2"/>
 <a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="prgcor_2.html#E9:19"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="prgcor_2.html#E4:19_2"><i><font color="Green" title="E23">A14</font></i></a>, <a class="txt" href="prgcor_2.html#E5:19_2"><i><font color="Green" title="E24">A15</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:19_2"/>
 <a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c8">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="prgcor_2.html#K4" title="PRGCOR_2:func.4">XFS2FS*</a> <font color="Maroon" title="c1">b</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>


</div>
